
LABORATORY 
ASTRONOMY 



WILLSON 




Class _^i>^ 

Book M-7- 

CopiglitN 



COPYRIGHT DEPOSIT 



LABORATORY ASTRONOMY 



BY 



ROBERT wVwiLLSON 



. 



BOSTON, U.S.A. 
GIKN" & COMPANY, PUBLISHERS 

1901 

V- 



THE LIBRARY Oh I 
CONGRESS, 

Two CoHits Received 

)UN. 15 1901 

Copyright entry 

CLASS a. X*e. N<*. 

COPY B. 



Copyright, 1900 
By ROBEKT W. WILLSON 



ALL RIGHTS RESERVED 






AS 



r* 



TABLE OF CONTENTS 

CHAPTER I 

THE SUN'S DIURNAL MOTION 

PAGE 

Path of the Shadow of a Pin-head cast by the Sun upon a Horizontal Plane 1 

Altitude and Bearing 4 

Representation of the Celestial Sphere upon a Spherical Surface ... 5 
The Sun's Diurnal Path upon the Hemisphere is a Circle — a Small Circle 

except about March 20 and September 21 8 

Determination of the Pole of the Circle 9 

Bearing of the Points of Sunrise and Sunset 11 

The Meridian — the Cardinal Points 11 

Magnetic Declination . . 12 

Azimuth 12 

The Equinoctial 14 

Position of the Pole as seen from Different Places of Observation . . 15 

Latitude equals Elevation of Pole 16 

Hour-angle of the Sun 17 

Uniform Increase of the Sun's Hour-angle — Apparent Solar Time . .18 

Declination of the Sun — its Daily Change 20 

CHAPTER II 

THE MOON'S PATH AMONG THE STARS 

Position of the Moon by its Configuration with Neighboring Stars . .21 

Plotting the Position of the Moon upon a Star Map .... 24 

Position of the Moon by Measures of Distance from Neighboring Stars . 25 

The Cross-staff 25 

Length of the Month . . .29 

Node of the Moon's Orbit 30 

Errors of the Cross-staff 31 

iii 



iv TABLE OF CONTENTS 

CHAPTER III 
THE DIURNAL. MOTION OF THE STARS 

PAGE 

Instrument for measuring Altitude and Azimuth 34 

Adjustment of the Altazimuth 35 

Determination of Meridian by Observations of the Sun .... 37 

Determination of Apparent Noon by Equal Altitudes of the Sun . . 39 

Meridian Mark 40 

Selection of Stars — Magnitudes . . .41 

Plotting Diurnal Paths of Stars on the Hemisphere .... 42 

Paths of Stars compared with that of the Sun 42 

Drawing of Hemisphere with its Circles 42 

Rotation of the Sphere as a Whole 43 

Declinations of Stars do not change like that of the Sun ... 43 

Equable Description of Hour-angle by Stars 43 

Hour-angle and Declination fix the Position of a Heavenly Body as well 
as Altitude and Azimuth — Comparison of the Two Systems of 

Coordinates 44 

Equatorial Instrument for measuring Hour-angle and Declination . . 45 

Universal Equatorial — Advantages of the Equatorial Mounting . . 45 

CHAPTER IV 

THE COMPLETE SPHERE OF THE HEAVENS 

Rotation of the Heavens about an Axis passing through the Pole explains 

Diurnal Motions of Sun, Moon, and Stars 47 

Relative Position of Two Stars determined by their Declinations and the 

Difference of their Hour-angles 48 

Use of Equatorial to determine Positions of Stars 49 

Use of a Timepiece to improve the Foregoing Method .... 50 
Map of Stars by Comparison with a Fundamental Star . . . .53 

Extension of Use of Timepiece to reduce Labor of Observation . . 54 

The Vernal Equinox to replace the Fundamental Star — Right Ascension 56 

Sidereal Time — Sidereal Clock . . . 57 

Right Ascension of a Star is the Sidereal Time of its Passage across the 

Meridian 58 

Right Ascension of any Body plus its Hour-angle at any Instant is Side- 
real Time at that Instant 58 

Finding Stars by the Use of a Sidereal Clock and the Circles of the Equa- 
torial Instrument . . . 59 

The Clock Correction , . 60 

List of Stars for determining Clock Error 61 



TABLE OF CONTENTS V 

CHAPTER V 
MOTION OF THE MOON AND SUN AMONG THE STARS 

PAGE 

Plotting Stars upon a Globe in their Proper Relative Positions . . 63 
Plotting Positions of the Moon upon Map and Globe by Observations of 
Declination, and Difference of Right Ascension from Neighboring 

Stars 64 

Variable Rate of Motion of the Moon 65 

Variable Semi-diameter of the Moon 65 

Position of Greatest Semi-diameter and of Greatest Angular Motion . 65 

Plotting Moon's Path on an Ecliptic Map 65 

Observations of Sun's Place in Reference to a Fundamental Star by Equa- 
torial and Sidereal Clock 66 

Sun's Place referred to Stars by Comparison with the Moon or Venus . 68 

Plotting the Sun's Path upon the Globe — the Ecliptic .... 70 

CHAPTER VI 
v 

MERIDIAN OBSERVATIONS 

Use of the Altazimuth or Equatorial in the Meridian 72 

The Meridian Circle 73 

Adjustments of the Meridian Circle 74 

Level 74 

Collimation 78 

Azimuth 78 

Determination of Declinations 80 

Determination of the Polar Point 81 

Absolute Determination of Declination 81 

Determination of the Equinox 83 

Absolute Right Ascensions 84 

Autumnal Equinox of 1899 85 

Autumnal Equinox of 1900 87 

Length of the Year . . . . .88 



PEEFATOEY NOTE 

The following pages are the beginning of an attempt to formulate a 
course of observation in preparation for the Laboratory Examination 
in Astronomy for admission to Harvard University. 

The first two chapters had been written out nearly in their present form 
as part of a proposed text-book for that purpose. The later portions, 
though less fully developed, are still in a form that may prove useful to 
exceptional students and certainly to teachers. 

In order to test the demand for such a presentation of the subject, it 
seemed worth while to print as much as is here given, for circulation 
among those who are interested in this method of teaching Astronomy, 
although the whole is not in suitable shape for use as a text-book. 

Chapters VII and VIII treat of the Nautical Almanac and the solution 
of problems by means of the celestial globe. 

These first eight chapters cover the subjects which are required of all 
candidates offering Astronomy for admission to Harvard University. The 
succeeding chapters will treat of those subjects from which selection may 
be made by each student in accordance with his tastes and opportunities. 

The method and apparatus here described have been used since 1897 in 
the summer course in Astronomy for teachers at Harvard, and in part 
in a few high schools. All the apparatus may be obtained from at least 
one of the firms supplying laboratory apparatus. 

Some copies of this edition have been interleaved in the binding for the 
convenience of teachers who may wish to make use of it in its present 
form, and in the hope of drawing suggestions from those interested. 



LABORATORY ASTRONOMY 



:>X*c 



Part I 



CHAPTER I 
THE DIURNAL MOTION OF THE SUN 

The most obvious and important astronomical phenomenon that 
men observe is the succession of day and night, and the motion of 
the sun which causes this succession is naturally the first object of 
astronomical study. Every one knows that the sun rises in the east 
and sets in the west, but very many educated people know little 
more of the course of the sun than this. The first task of the 
beginner in astronomy should be to observe, as carefully as possible, 
the motion of the sun for a day. What is to be observed then ? 
A little thought shows that it can only be the direction in which 
we have to look to see it at different times ; that is, toward what 
point of the compass — how far above the ground. All astronom- 
ical observation, indeed, comes down ultimately to this ■ — the direc- 
tion in which we see things. The strong light of the sun enables 
us to make use of a very simple method depending on the principle 
that the shadow of a body lies in the same straight line with the 
body and the source of light. 

Path of the Shadow of a Pin-head. — If we place a pin upright. on 
a horizontal plane in the sunlight and mark the position of the 
shadow of its head at any time, we thus fix the position of the 
sun at that time, since it is in the prolongation of the line drawn 
from the shadow to the pin-head. In order to carry out systematic 



2 LABORATORY ASTRONOMY 

observations by this method in such a form that the results may be 
easily discussed, it will be convenient to have the following appa- 
ratus : (1) A firm table in such a position as to receive sunlight for 
as long a period as possible. It is better that it should be in the 
open air, in which case it may be made by driving small posts into 




Fig. 1 

the ground and securely fastening a stout plank about 18 inches 
square as a top. (2) A board, 18 inches long and 8 inches broad, 
furnished with leveling screws and smoothly covered with white 
paper fastened down by (3) thumb tacks. (4) A level for leveling 
the board. (5) A compass. (6) A glass plate, 6 inches long and 
2 inches broad, along the median line of which a straight black 
line is drawn. (7) A pin, 5 cm. long, with a spherical head and 
an accurately turned base for setting it vertical. (8) A timepiece. 
Draw a straight pencil line across the center of the paper as 




Fig. 2 



nearly as possible perpendicular to the length of the board. Place 
the board upon the table and level approximately. Put the com- 
pass on the middle of the pencil line and put the glass plate on the 
compass with its central line over the center of the needle ; turn 
the plate till its median line is parallel to the pencil line (Fig. 2), 



TIIE DIUKNAL MOTION OF THE SUN 6 

and swing the whole board horizontally, till the needle is parallel to 
the two lines, which are then said to be in the magnetic meridian. 
Press the leveling screws firmly into the table, and thus make dents 
by which the board may at any future time be placed in the same 
position without the renewed use of the compass. Level the board 




Fig. 3 



carefully, placing the level first east and west, then north and south. 
Place the pin in the pencil line, — in the center if the observation 
is made between March 20 and September 20, but near the south- 
ern edge of the board at any other time of the year, — pressing it 
firmly down till the base is close to the paper, so that the pin is 
perpendicular to the paper. Mark with a hard pencil the estimated 
center of the shadow of the pin-head, A (Fig. 3), noting the time by 
the watch to the nearest minute, affix a number or letter, and affix 
the same number to the recorded time of the observation in the 
note-book. It is a good plan to use pencil for notes made while 




fig. 4 



observing, and ink for computations or notes added afterward in 
discussing them. Repeat at hourly, or better half-hourly, intervals, 
thus fixing a set of points (Pig. 4), through which a continuous 
curve may be drawn showing the path of the shadow for several 
hours. The same observation should be repeated two weeks later. 



LABORATORY ASTRONOMY 



ALTITUDE AND BEARING 



By the foregoing process we obtain a diagram on which is shown 
the position of the pin point, a magnetic meridian line through this 
point, and a series of numbered points showing the position of the 
shadow of the pin-head at different times ; the height of the pin is 
known and also the fact that its head was in the same vertical line 
with its point. 

In the discussion of these results, it will be convenient to proceed 
as follows : 

Kemove the pin and draw with a hard pencil a fine line, AB 
(Fig. 5), through the pinhole and the point marked at the first obser- 
vation. This line is called a line of bearing, and the angle which 




Fig. 5 



it makes with the magnetic meridian is called the magnetic Rearing 
of the line. This angle, which may be directly measured on the 
diagram by a protractor, fixes the position of the vertical plane which 
contains the observed point and passes also through the center of 
the pin-head and the sun. If this point bears KW. from the pin, 
the sun evidently bears S.E. 

Imagine a line, A C (Fig. 3), connecting the observed point with the 
sun's center and passing also through the center of the pin-head. 
The position of the sun in the vertical plane is evidently fixed by 
this line. The angle between the line of bearing and this line, BA C, 
is called the altitude of the sun ; it measures, by the ordinary con- 
vention of solid geometry, the angle between the sun's direction 
and the plane of the horizon. 



THE DIURNAL MOTION OP 1 THE SUN 5 

To determine this angle, lay off the line B'C (Fig. 6), equal in 
length to the pin, 5 cm., draw a perpendicular through B' ; and by 
means of a pair of compasses or scale laid 
between the two points A and B (Fig. 5), 
lay off the line A'B' on the perpendicular, 
draw A'C, and measure the angle B'A'C 
by a protractor. We now have the bearing 
and altitude of the sun at the time of the 
first observation, the bearing of the sun 
from the pin being opposite to that of 
the point from the pin. , In this manner 
the altitude and bearing are determined for 
each observed point upon the path of the 
shadow, and noted against the correspond- 
ing time, in the note-book (to avoid con- 
fusion, it is convenient to«make a separate 
figure for the morning and afternoon 
observations, as shown in Fig. 6). We 
have thus obtained a series of values 
which will enable us to study more easily 
the path of the sun upon the concave of 
the sky. 

Plotting the Sun's Path on a Spherical Surface. — Probably the 
most evident method of accomplishing this object would be to 





Fig. 7 



construct a small concave portion of a sphere, as in the accom- 
panying figure, which suggests how the position of the sun might 
be referred to the inside of a glass shell. 



LABORATOKY ASTRONOMY 



But the hollow surface offers difficulty in construction and 
manipulation, and it requires but little stretch of the imagination 
to pass to the convex surface as follows. The glass shell, as 
seen from the other side, would appear thus : 




Fig. 8 



and we can more readily get at it to measure it, and moreover can 
more easily recognize the properties of the lines which we shall 
come to draw upon it, since we are used to looking upon spheres 
from the outside rather than from the inside, except in the case of 
the celestial sphere. 

On both Figs. 7 and 8 is shown a group of dots which have 
nearly the configuration of a group of stars conspicuous in the 
southern heavens in midsummer and called the constellation of 
Scorpio. It is evident that the constellation has the same shape in 
both cases, except that in Tig. 8 it is turned right for left or semi- 
inverted, as is the image of an object seen in a mirror. This prop- 
erty obviously belongs to all figures drawn on the concave surface 
as seen from the center, when they are looked at from the outside 
directly toward the center. 

So also the diurnal motion of the sun, which as we see it 
from the center is from left to right, would be from right to 
left as viewed from the outside of such a surface. This latter 
is so slight an inconvenience that it is customary to represent 
the motions of the heavenly bodies in the sky upon an opaque 
globe, and to determine the angles which these bodies describe 
about the center, by measuring the corresponding arcs upon the 
convex surface. 



THE DIURNAL MOTION OF THE SUN 



Plotting on a Hemisphere. — The apparatus required for plotting 
the sun's path consists of : a hemisphere, a, 4J- inches in diameter ; 
a circular protractor, b, a quadrantal protractor, c, of 2£ inches 






Fig. 9 



radius, and a pair of compasses, d, whose legs may be bent and one 
of which carries a hard pencil point. 

Determine by trial with, the compasses the center of the base of 
the hemisphere, and mark two diameters by drawing straight lines 
upon the base at right angles through the center. Prolong these by 
marks about -J inch in length upon the convex surface. Place the 




Fig. 10 



hemisphere exactly central upon the circular protractor, by bring- 
ing the marked ends of one of the diameters upon those divisions 
of the protractor which are numbered 0° and 180°, and the other on 
the divisions numbered 90° and 270°. Determine and mark the 



8 



LABORATORY ASTRONOMY 



highest point of the hemisphere by placing the quadrant with its 
base upon the circular protractor, and its arc closely against the 
sphere, and marking the end of the scale (Fig. 10). Eepeat this 
with the arc in four positions, 90° apart on the base. The points 
thus determined should coincide ; if they do not, estimate and mark 
the center of the four points thus obtained. This point represents 
the highest point of the dome of the heavens — the point directly 
overhead, called the zenith, and the zero and 180° points on the base 
protractor may be taken as representing the south and north points 
respectively of the magnetic meridian. 

The Sun's Path a Circle. — To plot the altitude and bearing of the 
first observation, place the foot of the quadrant or altitude arc 
close against the sphere, the foot of its graduated face on the 
degree of the protractor which corresponds to the bearing. Mark 
a fine point on the sphere at that degree of the altitude arc corre- 
sponding to the altitude at the first observation. This point fixes 
the direction in which the sun would have been seen from the center 
of the hemisphere at the time of observation if the zero line had 
been truly in the magnetic meridian. Proceed in the same manner 
with the other observations of bearing and altitude, and thus obtain 




Fig. 11 



a series of points (Fig. 11), through which may be drawn a con- 
tinuous line representiDg the sun's path upon that day. 

It will appear at once that the arcs between the successive points 
are of nearly equal length if the times of observation were equi- 
distant, and otherwise are proportional to the intervals of time 



THE DIURNAL MOTION OF THE SUN V 

between the corresponding observations — a property which does 
not at all belong to the shadow curve from which the points are 
derived. We thus have a noteworthy simplification in referring 
our observations to the sphere. It will also appear that a sheet of 




Fig. 12 



stiff paper or cardboard may be held edgewise between the hemi- 
sphere and the eye, so as to cover all the points ; that is, they all 
lie in the same plane. This fact shows that the sun's path is a 
circle on the sphere. It is shown by the principles of solid geometry 
that all sections of the sphere by a plane are circles. If the plane 
of the circle passes through the center, it is the largest possible, its 
radius being equal to that of the sphere ; it is then called a great 
circle. Near the 20th of March and 22d of September it will be 
found that the path of the shadow is nearly a straight line on the 
diagram, and that the path of the sun is nearly a great circle ; that 
is, the plane of this circle passes nearly through the center of the 
sphere. In general, the shadow path is a curve, with its concave 
side toward the pin in summer and its convex side toward it in 
winter, while the path on the sphere is a small circle, that is, its 
plane does not pass through the center of the sphere. 

Determining the Pole of the Circle. — It is proved by solid geometry 
that all points of any circle on the sphere are equidistant from two 



10 



LABORATORY ASTRONOMY 



points on the sphere, called the poles of the circle. It is important 
to determine the pole of the sun's diurnal path. 

Estimate as closely as possible the position on the sphere of a 
point which is at the same distance from all the observed points of 
the sun's path and open the compasses to nearly this distance. For 
a closer approximation to the position of the pole, place the steel 
point of the compasses at the point on the hemisphere correspond- 
ing to the first observation, a, and with the other (pencil) point draw 
a short arc, m (Fig. 12), near the estimated pole. Draw the arc n 

from the point of the 
last observation, e, and 
join these two arcs by 
a third drawn from an 
observed point, b, as 
near as possible to the 
middle of the path; 
the pole of the sun's 
diurnal circle will lie 
nearly on the great 
circle drawn from b to 
the middle point o of 
the arc last drawn. 
Place the steel point 
at o, and the pencil 
point at b, and try the 
distance of the pencil 
point from the sun's 
, ., T „' ., path at either ex- 

tremity. If the pencil point lies above (or below) the path at both 
extremities the compasses must be opened (or closed) slightly and 
the assumed pole shifted directly away from (or toward) the middle 
of the path. ' 

The proper opening of the compasses is thns quickly determined 
as well as a close approximation to the position of the pole Place 
the steel point at this new position,^, the pencil point at b, and 
again test the extreme points. If the west end of the path is below 
the pencil point (Fig. 13), the latter should be brought directly down 




Fig. 13 



THE DIURNAL MOTION OF THE SUN 11 

to the path by shifting the steel point on the sphere in the plane 
of the compass legs, that is, along the great circle from p to s. 

From the point thus found a circle can be described with the 
compasses so as to pass approximately through all the observed 
points ; that is, this point is the pole of the sun's path, and when 
it is fixed as exactly as possible a circle is to be drawn from horizon 
to horizon which will represent the sun's path from the point of 
sunrise to that of sunset, and passing very nearly through all the 
observed points. The bearing of the points of sunrise and sunset 
may then be read off on the horizontal circle. 

THE MERIDIAN 

The pole as thus determined marks a very interesting and 
important point in the heavens. We will draw a great circle 
through the zenith and the pole. To do this, place the altitude arc 
against the sphere, as if to measure the altitude of the pole ; and 




Fig. 14 

using it as a guide, draw the northern quadrant of the vertical 
circle through the zenith and the pole. Note the bearing of this 
vertical circle. Place the altitude arc at the opposite bearing, and 
draw another or southern quadrant of the same great circle till it 
meets the south horizon. This great circle (Fig. 14) is called the 
meridian of the place of observation, and its plane is called the 
plane of the meridian of the place of observation, — sometimes 
the true meridian, to distinguish it from the magnetic meridian. 



12 LABOKATOKY ASTRONOMY 

The line in which it cuts the base of the hemisphere represents the 
meridian line or true meridian line, just as the line first drawn repre- 
sents the line of the magnetic meridian. If the observations are made 
in the United States, near a line drawn from Detroit to Savannah, 
it will be found that the true meridian coincides very nearly with 
the magnetic meridian. East of the line joining these cities, the 
north end of the magnet points to the west of the true meridian by 
the amounts given in the following table : 

21° at the extreme N.E. boundary of Maine. 
15 at Portland. 
10 at Albany and New Haven. 
5 at Washington and Buffalo. 

While on the west the declination, as it is called, is to the east of the 
true meridian. 

5° at St. Louis and New Orleans. 
10 at Omaha and El Paso. 
15 at Dead wood and Los Angeles. 
20 at Helena, Montana, and C. Blanco. 
23 at the extreme N.W. boundary of the United States. 

By drawing these lines on the map, as in Fig. 15, it is easy to 
estimate the declinations at intermediate points within one or two 
degrees, — at the present time west declinations in the United States 
are increasing and east declinations decreasing by about 1° in fifteen 
years. 

A great circle perpendicular to the meridian may be drawn by 
placing the altitude protractor at readings 90° and 270° from the 
meridian reading and drawing arcs to the zenith in each case. 
This circle is the prime vertical, and intersects the horizon in the 
east and west points ; thus all the cardinal points are fixed by the 
meridian determined from our plotting of the sun's path. 

Azimuth. — Place the hemisphere upon the circular protractor in 
such a position that the line of the true meridian on the hemisphere 
coincides with the zero line of the protractor. 

Place the altitude arc so as to measure the altitude at any part of 
the sun's path west of the meridian (Fig. 16). The reading of the 
foot of the arc will give the angle between the true meridian and 



THE DIURNAL MOTION OF THE SUN 



13 



the vertical plane containing the sun at that point of its diurnal 
circle. This angle is its true bearing and differs from its magnetic 




Fig. 15 



bearing by the declination of the compass, being evidently less than 
the magnetic bearing, if the declination is west of north. It is also 
called the azimuth of the sun's vertical circle, or, briefly, of the sun. 




Fig. 16 



Formerly azimuth was usually reckoned from north through the 
west or east, to 180° at the south point. It is now customary to 
measure it from south through west up to 360°, so that the azimuth 



14 LABORATORY ASTRONOMY 

of a body when east of the meridian lies between 180° and 360°. 
The present method is more convenient because the given angle 
fixes the position of the vertical circle without the addition of the 
letters E. and W. It is worthy of notice that with this notation 
the azimuth of the sun as seen in northern latitudes outside of the 
tropics always increases with the time ; and indeed this is true of 
most of the bodies we shall have occasion to observe. 

Now place the altitude quadrant so that its foot is at a point on 
the circular protractor where the reading is 360° minus the azimuth 
of the point just measured ; the sun at this point of its path is just as 
far east of the meridian as it was west of the meridian at the point 
last considered, and it will be found that the altitude of the two 
points is the same. On the path shown in Fig. 16 the altitude is 
45° at the points whose azimuths are 60° and 300° (60 E. of S.). 

This fact, that equal altitudes of the sun correspond to equal 
azimuths east and west of the true meridian, is an important one, 
and will presently be made use of to enable us to determine the 
position of the true meridian with a greater degree of precision. 

THE EQUINOCTIAL 

We shall find it convenient to draw upon the hemisphere another 
line, which plays an important role in astronomy, the great circle 
90° from the pole. Placing the steel point of the compasses at 
the zenith, open the legs until the pencil point just comes to the 
horizon plane where the spherical surface meets it, so that if it 
were revolved about the zenith, the pencil point would move in 
the horizon. The compass points now span an arc of 90° upon the 
hemisphere. Place the steel point at the pole, and draw as much 
of a great circle as can be described on the sphere above the horizon. 
This will be just one-half of the great circle, and will cut the horizon 
in the east and west points. The new circle is called the equinoctial 
or celestial equator (Eig. 17). 

We have seen that the path of the sun over the dome of the 
•heavens appears to be a small circle described from east to west 
about a fixed point in the dome as a pole. The ancient explanation 
of this fact was that the sun is fixed in a transparent spherical shell 



THE DIURNAL MOTION OF THE SUN 



15 



of immense size revolving daily about an axis, the earth being a 
plane in the center of unknown extent, but whose known regions 
are so small compared to the shell that from points even widely 
separated on the earth the appearance is the same ; just as the 




Fig. 17 

apparent direction and motion of the sun would be practically 
the same on our hemisphere to a microscopic observer at the 
center, and to another anywhere within one-hundredth of an inch 
of the center. When observations were made, however, at points 
some hundreds of miles apart on the same meridian, very per- 
ceptible differences were found, whose nature will be understood 
from a comparison of the hemisphere (Fig. 18 a), plotted from 




Fig. 18 



observations made Aug. 8, 1897, at a point in Canada, not far 
from Quebec, with a second hemisphere (Fig. lSb), on which is 
shown the path of the sun on the same date derived from observa- 
tion of the shadow of a pin-head at Polf os in Norway. It appears 
on comparison that the distance of the pole above the north horizon 



16 LABOKATOKY ASTKONOMY 

is considerably greater in the latter, while the equator is just as 
much nearer the southern horizon ; the sun is at the same distance 
from the equator in each case. This fact cannot be explained on 
the supposition that the horizon planes of the two places are the 
same, for in that case we should have the spherical shell which 
contains the sun revolving at the same time about two different 
fixed axes, which is impossible. It is not, however, improbable 
that the earth's surface should be curved, if we can admit as 
a possibility that the direction of gravity, which is perpendicular 
to a horizontal plane, may be different at different places. That 
the earth's surface in the east and west direction is curved, we 
know; for men have traversed it from east to west and returned 
to the starting point, so that we have good reason to believe that its 
surface is everywhere curved. Long before this conclusive proof 
was obtained, however, the globular form of the earth was inferred 
on good grounds. 

It was early suggested (regarding the fact that, if the sun is fixed 
in a shell, that shell is of enormous size as compared with the earth) 
that it is inherently more probable that the apparent motion of 
the sun is due to a rotation of the spherical earth about an axis 
passing through the earth's center and the poles of the sun's circle. 
This argument is greatly strengthened when we investigate the 
apparent motion of the stars in connection with their size and dis- 
tance, and it is now beyond a doubt that this is the true explanation 
of the apparent diurnal motion of the sun. 



LATITUDE EQUALS ELEVATION OF THE POLE 

This subject is treated in all text-books on descriptive astronomy, 
and it is pointed out that the pole of the sun's path is the point 
where the line of the earth's axis of rotation cuts the sky, and the 
equinoctial or celestial equator is the great circle in which the plane 
of the earth's equator cuts the sky. The fact is proved also that 
the elevation of the pole above the horizon at any place is equal to 
the latitude of the place. 

This angle, as measured on the hemisphere shown in Fig. 18 a, is 
47°, and on the hemisphere of Fig. 18 b is 62°. The latitudes of 



THE DIURNAL MOTION OF THE SUN 



17 



Quebec and Polfos as determined by more accurate measures are 
46° 50' and 61° 57'. 

It is easy to see that the arc of the meridian from the zenith 
to the equinoctial is also equal to the latitude, while the arc from 
the south point of the horizon to the equator and that from the 
zenith to the pole are each equal to 90° minus the latitude, or, as it 
is usually called, the co-latitude. 

It will be well here, as in all our measurements, to form some idea 
of the accuracy of our results. As one degree on our hemisphere 
is quite exactly equal to l mm , a quantity easily measured by ordi- 
nary means, it is not difficult with ordinary care to determine the 




Fig. 19 

pole of the sun's path so closely that no observed point lies more 
than a degree from the path. The pole is then fixed within one 
degree unless the length of the path is very short ; usually if the 
path is more than 90° in length the pole may be placed within less 
than a degree of its true place and the latitude measured with an 
error of less than one degree. 



HOUR-ANGLE OF THE SUN 

Open the dividers as before (see p. 14) so as to draw a great circle. 
Place the steel point upon the place of the sun, S, on its diurnal 
circle at the time of the last observation in the afternoon (Fig. 19), 
and with the pencil point strike a small arc cutting the equator at Q. 



18 LABORATORY ASTRONOMY 

Place the steel point where this arc cuts the equator, and draw a 
great circle which will pass through the sun's place and the pole ; 
notice that it also cuts the equator at right angles. Such a circle is 
called an hour-circle. It is the intersection of the surface of the 
sphere with a plane that passes through the poles and the place of 
the sun. The number of degrees in the arc of the equator, included 
between the meridian and the hour-circle which passes through the 
sun, is called the hour-angle of the sun. By the ordinary convention 
of solid geometry it measures the wedge angle between the plane of 
the hour-circle and the plane of the meridian. If a book be placed 
with its back in the line from the pole to the center of the sphere, 
and with its title-page to the west, and the western cover opened 
till it is in the plane of the hour-circle, while the title-page is in 
the plane of the meridian, the wedge angle between the title-page 
and the cover will be the hour-angle and will be measured by the 
arc of the equator indicated above. It is reckoned as increasing 
from the meridian towards the west in the direction in which the 
cover is opened. If the hour-circle of the first morning observa- 
tion is determined in the same way, the hour-angle measured in 
the opposite direction from the meridian is sometimes called the 
hour-angle east of the meridian ; but more commonly by astronomers 
this value is subtracted from 360°, and the angle thus obtained is 
called the hour-angle, this being more convenient because the hour- 
angle of the sun thus measured constantly increases with the time 
as the sun pursues its course ; being 0° at noon, 180° at midnight, 
360° at the next noon, etc. 

UNIFORM INCREASE OF HOUR-ANGLE 

Let us now examine more carefully the truth of the surmise pre- 
viously made, that the arc of the sun's path between two successive 
observations is proportional to the interval of time between the 
observations. Draw the hour-circles of the sun at each point of 
observation (Fig. 20) ; measure the arc on the equator between the 
first and the last hour-circles ; divide by the number of minutes 
between the two times. This will give the average increase of 
hour-angle per minute. Multiply this increase by the difference in 



THE DIUKNAL MOTION OF THE SUN 



19 



minutes of each of the observed times from the time of the first 
observation, and compare with the progressive increase of the hour- 
angle as measured off on the equator by means of the graduated 
quadrant. They will be found to be nearly the same in each case. 
It is thus shown that the hour-angle of the sun increases uniformly 
with the time. The rate is nearly a quarter of a degree per minute, 
since 360° are described in 24 hours. Notice that when the hour- 
angle is zero, the actual time by the watch is not very far from 12 
o'clock (in extreme cases it may be 45 minutes, if the clock is keep- 
ing standard time), and that if the hour-angle in degrees (west of 
the meridian) is divided by 15, the number of hours differs from the 




Fig. 20 



watch time just as much as the time of meridian passage differs 
from 12 hours. In fact, the hour-angle of the sun measures what 
is called apparent solar time, i.e., when H.A. = 15°, it is 1 o'clock; 
H. A. = 75°, it is 5 o'clock ; H.A. = 150°, 10 o'clock, etc. ; those angles 
east of the meridian lying between 180° and 360°, i.e., between 12 h 
and 24 h , so that 12 hours must be subtracted to give the correct hours 
by the ordinary clock, which divides the day into two periods of 24 
hours each ; for instance, if H.A. = 270°, it is 18 h past noon or 6 a.m. 
of the next day. Astronomical clocks usually show the hours con- 
tinuously from to 24, thus avoiding the necessity of using a.m. 
and p.m. to discriminate the period from noon to midnight and from 
midnight to noon. 



20 LABOKATOKY ASTKONOMY 



DECLINATION OF THE SUN 



The distance of the sun's path from the celestial equator, meas- 
ured along the arc of an hour-circle, is called its declination, and 
will be found appreciably the same at all points. It requires more 
delicate observation than ours to find that it changes during the 
few hours covered by our observation. If, however, the observa- 
tion be repeated after an interval, say, of two weeks at any time 
except for a month before or after the 20th of June or December, 
it will be found that although the sun at the second observation 
describes a circle, this circle is not in the same position with regard 
to the equator — that its declination has changed (between March 
13 and 27, for instance, by about 5°.5). The inference to be drawn 
is that even during the period of our observation the sun's path is 
not exactly parallel to the equator, although our observations are 
not delicate enough to show that fact. 

It is true in general, as in this case, that the first rude meas- 
urements applied to the heavenly bodies give results which when 
tested by those covering a longer time, or made with more delicate 
instruments, are found to require correction. 



CHAPTER II 
THE MOON'S PATH AMONG THE STARS 

Next to the diurnal motion of the sun the most conspicuous 
phenomenon is the similar motion of the stars and the moon ; this 
will form the subject of a future chapter. 

The study of the moon, however, discloses a new and interesting 
motion of that body. It partakes indeed of the daily motion of 
the heavenly bodies from east to west, but it moves less rapidly, 
requiring nearly 25 hours to complete its circuit instead of 24, as 
do the sun and stars, and returning to the meridian therefore 
about an hour later on each successive night. 

In consequence of this motion it continually changes its place 
with reference to the stars, moving toward the east among them 
so rapidly that the observation of a few hours is sufficient to show 
the fact. At the same time its declination changes like that of the 
sun, but much more rapidly. 

We should begin early to study this motion, and it will be found 
interesting to continue it at least for some months at the same time 
that other observations are in progress — a very few minutes each 
evening will give in the course of time valuable results. 

POSITION BY ALIGNMENT WITH STARS 

The first method to be used consists in noting the moon's place 
with reference to neighboring stars at different times. Some sort 
of star map is necessary upon which the places of the moon may be 
laid down so that its path among the stars may be studied. As 
the configurations that offer themselves at different times are of 
great variety, it will be well to give a few examples of actual 
observations of the moon's place by this method. 

Dec. 12, 1899, at 12 h m p.m., the moon was seen to be near 
three unknown stars, making with them the following configuration, 

21 



22 LABORATORY ASTRONOMY 

which was noted on a slip of paper as shown in Fig. 21. The 
relative size of the stars is indicated by the size of the dots. (The 
original papers on which the observations are made should be care- 
ts* Dec i2* Mty preserved ; indeed, this should always be the 
practice in all observations.) 
****.^ At the same time, for purposes of identification, 
/ it was noted that the group of stars formed, with 
/ Capella and the brightest star in Orion, both of 
° M . •- which were known to the observer, a nearly equi- 

A symmetrical figure J x 

lateral triangle. It was also noted that the moon 
was about 6° from the farthest star, this being 
estimated by comparison with the known distance between the 
" pointers' 7 in the "Dipper" (about 5°). With these data it was 
easily found by the map that these stars were the brightest stars 
in Aries, and the moon was plotted in its proper place on the map 
(page 24). 

December 13, at 5 h 35 m p.m., the moon was £° (half its diameter) 
below (south of) a line drawn from Aldebaran (identified by its 
position with reference to Capella and Orion and by the letter V of 
stars in which it lies, the Hyades) to the faintest of the three 
reference stars of December 12. It was also about f° west of a 
line between two unknown stars identified later as Algol (equi- 
distant from Capella and Aldebaran) and y Ceti (at first supposed 
on reference to the map to be a Ceti, D ^ a a 3 h 35 m 
but afterward correctly identified by \ ~ 
comparing the map with the heavens). '• 
The original observation is given \ ^^ 

below (Fig. 22) of about one-half the % \ '■ . fi 

size of the drawing, all except the -.- ; «-— -— — - yo 

., , , . . ., aldebaran \ 

underscored names being in pencil. •. 

The underscored names are in ink and e. \ 

made after the stars were identified. 
This is a useful practice when addi- 
tions are made to an original, so that subsequent work may not be 
given the appearance of notes made at the time of observation. It 
is well to give on the sketch map several stars in the neighborhood 
of those used for alignment, to facilitate identification. 



THE MOON'S PATH AMONG THE STARS 23 

The alignment was tested by holding a straight stick at arm's 
length parallel to the line joining the stars. 

December 14, 6 h 30 m p.m. Moon on a line from Algol through 
the Pleiades (known) about 2\° (5 diameters of moon) beyond the 
latter, which were very faint in pg^.^ . 'capeiia Dec ws^io™ 

the strong moonlight. No figure. 

December 15, 5 h 10 m p.m. Moon \ \ 

in a line between Capella and Aide- \ \ 

baran. Line from Pleiades to moon flT . . \ • 
bisects line from Aldebaran to /? \ \ 

Tauri (identified by relation to — -V-6 A 

Aldebaran and Capella). \\ 

9 h 25 m p.m. Moon in line from \ . 

P Aurigae to Aldebaran (Fig. 23). ""^Tas ' ' 

(Note. — Henceforth details of identification are omitted.) 

December 16, 7 h 40 m p.m. Moon almost totally eclipsed 2\° east 
of line from fi Aurigae to y Orionis ; same distance from yS Tauri as 
^ Tauri (revised estimate about £° nearer f$ Tauri 
°^» than is £ Tauri) (Fig. 24). 

December 18, 10 h 30 m p.m. Observation 
snatched between clouds. Moon's western edge 
tangent to. line from a G-eminorum to Procyon 
and about 1° north of center of that line. 

In the sketch maps above no great accuracy is 

attempted in placing the stars, but in the final 

plotting on the map the directions of the notes 

are carefully followed. The plotting should be 

done as soon as possible after the observation 

is made, for even a hasty comparison with the 

map will often show that stars have been mis- 

identified or that there is some obvious error in 

fig. 24 the notes, which may be rectified at once if there 

is an opportunity to repeat the observation. Such 

a case occurs in the observations of December 13 recorded above, 

where y Ceti was mistaken for a. 



24 



LABORATORY ASTRONOMY 



PLOTTING POSITIONS OF THE MOON ON A STAR MAP 

Figure 25 shows the positions of the moon plotted from the fore- 
going observations, together with the lines of construction from 
which they were determined. 

A drawing should be made of the shape of the illuminated portion 
of the moon at each observation, and the direction among the stars 







# » 


• 






• 


• 


• 


* • 


\« 


• 


\ 




• • . 




K\ 


\ '>is 




• 










"°^\~~ — : ^-^; 






,* 


\ ■*'■ 


V 1 ;" 


k7r ^ 


-_.'' 13 


.12 * 
>-- 


r 




• ' ; 






•' \ 








t 

» • 

• *• 























' 



Fig. 25 



of the line joining the points of the horns (cusps) for future study 
of the cause of the moon's changes of phase. 

If the star map accompanying this book is used, the identification 
of the stars consists in determining which of the dots represents 
the star of reference; the name may be determined by reference 
to the list ; thus the two stars near the line XXIV on the upper 
portion of the map are " a Andromedse h 5 m + 29° " and " y Pegasi 
h 8 m + 14°." The meaning which attaches to these numbers is 
given in Chapter III. It is a good plan to keep a copy of the 
map on which to note the names for reference as the stars are 
learned; most of the conspicuous ones will soon be remembered 
as they are used. 



THE MOONS PATH AMONG THE STARS 



25 



THE MOON'S PLACE FIXED BY ITS DISTANCE FROM 
NEIGHBORING STARS 

One month's observation by this method will show that the moon's 
path is at all points near to the curved line drawn on the map, 
which is called the ecliptic and which is explained on page 70. 
To establish more accurately its relations to this line it will be 
advisable in the later months to adopt a more accurate means of 
observation, although when the moon is very near a bright star, its 
position may be quite accurately fixed by the means that we have 
indicated ; and if it chances to pass in front of a bright star and 
produce an occupation, the moon's position is very accurately fixed 
indeed, as accurately as by any method. But such opportunities 
are rare, and for continuous accurate observation we should have a 
means of measuring the distance of the moon from stars that are 
at a considerable distance from it. An instrument sufficiently accu- 
rate for our purpose is the cross-staff described below. It should be 
mentioned that, on ac- 
count of the distortion 
of the map, the place of 
the moon is usually more 
accurately given by dis- 
tances from the com- 
parison stars than by 
alignment. The sextant 
may be used instead of 
the cross-staff, but is 
less convenient and also 
more accurate than is 
necessary. 

The Cross-staff. — The 
cross-staff (Fig. 26) con- 
sists of a straight graduated rod upon which slides a "transversal" 
or " cross " perpendicular to the rod ; one end of the staff is placed 
at the eye and the " cross " is moved to such a place that it just 
fills the angle from one object to another; its length is then the 
chord of an arc equal to the angle between the objects as seen from 




Fig. 26 



26 LABORATORY ASTRONOMY 

that end of the staff at which the eye is placed. The figure, which 
is taken from an old book on navigation, illustrates the use of this 
instrument for measuring the sun's altitude above the sea horizon ; 
the rod in the position shown indicates that the sun's altitude is 
about 40°. 

Obviously a given position of the cross corresponds to a definite 
angle at the end of the rod, and the rod may be graduated to give 
this angle directly by inspection, or a table may be constructed by 
which the angle corresponding to any division of the rod may be 
found ; such a table is given on page 27. For our purpose an instru- 
ment of convenient dimensions is made by using a cross 20 cm. in 
length, sliding on a rod divided into millimeters (Fig. 27) ; this may 
be used for measuring angles up to 30°, which is enough for our 



£ 



ll}lililtMltlil!Jliil}|}ltlM.M'.!?ilt.' 



t 



IWWB I 'l'l I iM ' HHH 'I U 'I'II H. I M ' H ' ' LM ' l ' ' ' ' 'I M I| M '|i UHM'IM 'l ') i | ' M,l i| I 'l' l 'l'l ' l 



FIG. 27 



purpose. The smallest angle that can be measured is about 12°, 
which corresponds to a chord of £ of the radius ; but by making a 
part of the cross only 10 cm. long, as shown in the figure, we may 
measure angles from 6° upwards, and for smaller angles may use 
the thickness of the cross, which is 5 cm., and thus measure angles 
as small as 3°; the longer cross will not give good results above 
&0°, as a slight variation of the eye from the exact end of the rod 
makes a perceptible difference in the value of the angles greater 
than 30°. 

Measures with the Cross-staff. — As an example of the use of the 
cross-staff, the following observations are given: They were made 
with a staff about 3 feet in length, graduated by marking the point 
for each degree at the proper distance in millimeters from the eye 
end of the staff, as given by Table II on page 27. After the points 
were marked a straight line was drawn through each entirely across 
the rod, using the cross itself as a ruler; graduations were thus 
made on one side for use with the 20 cm. cross, on the other for the 



THE MOON S PATH AMONG THE STARS 



27 



Table I - 


— Angle subtended by 


Crosses 


Table II 


Distance 


Length of Cross 


Distance 


Length of Cross 


Angle 


from 








from 








subtended by 


Eye 


20 cm. 


10 cm. 


5 cm. 


Eye 


20 cm. 


10 cm. 


5 cm. 


20cir 


. Cross 


100 c,n 


11°. 4 


5°. 7 


2°. 9 


62 cm 


18°. 3 


9°. 2 


4°. 6 






99 


11 .5 


5 .8 


2 .9 


61 


18.6 


9.4 


4.7 






98 


11 .6 


5.8 


2 .9 


60 


18 .9 


9.5 


4.8 






97 


11 .8 


5.9 


3.0 


59 


19 .2 


9.7 


4 .9 






96 


11 .9 


6 .0 


3.0 


58 


19.6 


9.9 


4.9 






95 


12 .0 


6.0 


3 .0 


57 


19.9 


10.0 


5.0 






94 


12.1 


6.1 


3.0 


56 


20 .2 


10 .2 


5 .0 


12° 


951 mm 


93 


12 .3 


6.2 


3.1 


55 


20.6 


10.4 


5 .2 


13 


878 


92 


12.4 


6.2 


3.1 


54 


21 .0 


10 .6 


5.3 


14 


814 


91 


12.5 


6.3 


3.1 


53 


21 .4 


10.8 


5 .4 


15 


760 


90 


12 .7 


6 .4 


3 .2 


52 


21 .8 


11 .0 


5 .5 


16 


711 


89 


12 .8 


6 .4 


3.2 


51 


22 .2 


11 .2 


5.6 


17 


669 


88 


13 .0 


6 .5 


3.3 


50 


22 .6 


11 .4 


5 .7 


18 


631 


87 


13.1 


6 .6 


3.3 


49 


23.1 


11 .6 


5.8 


19 


598 


86 


13.3 


6.7 


3.3 


48 


23 .5 


11 .9 


6 .0 


20 


567 


85 


13.4 


6 .7 


3.4 


47 


24 .0 


12.1 


6.1 


21 


540 


84 


13.6 


6.8 


3.4 


46 


24 .5 


12 .4 


6 .2 


22 


514 


83 


13.7 


6.9 


3.5 


45 


25 .1 


12 .7 


6.4 


23 


491 


82 


13 .9 


7 .0 


3.5 


44 


25.6 


13.0 


6.5 


24 


470 


81 


14.1 


7 .1 


3.5 


43 


26 .2 


13.3 


6.7 


25 


451 


80 


14.3 


7 .2 


3.6 


42 


26.8 


13.6 


6 .8 


26 


433 


79 


14 .4 


7 .2 


3.6 


41 


27 .4 


13 .9 


7 .0 


27 


416 


78 


14.6 


7 .3 


3.7 


40 


28 .1 


14 .3 


7 .2 


28 


401 


77 


14 .8 


7 .4 


3.7 


39 


28.8 


14 .6 


7 .3 


29 


387 


76 


15 .0 


7 .5 


3.8 


38 


29 .5 


15 .0 


7 .5 


30 


373 


75 


15 .2 


7 .6 


3.8 


37 


30.2 


15 .4 


7 .7 


31 


361 


74 


15.4 


7 .7 


3.9 


36 


31 .0 


15.8 


7 .9 


32 


349 


73 


15 .6 


7 .8 


3.9 


35 


31.9 


16.3 


8.2 


33 


338 


72 


15.8 


7.9 


4.0 


34 


32 .8 


16.7 


8.4 


34 


327 


71 


16 .0 


8.1 


4.0 


33 


33.7 


17 .2 


8.7 


35 


317 


70 


16 .3 


8.2 


4.1 


32 


34.7 


17 .7 


8.9 


36 


308 


69 


16 .5 


8.3 


4.2 


31 


35.8 


18.3 


9.2 


37 


299 


68 


16 .7 


8.4 


4.2 


30 


36 .9 


18 .9 


9.5 


38 


290 


67 


17 .0 


8.5 


4.3 


29 


38.1 


19.6 


9.9 


39 


282 


66 


17 .2 


8.7 


4.3 


28 


39.3 


20.2 


10 .2 


40 


275 


65 


17 .5 


8.8 


4.4 


27 


40 .6 


21 .0 


10 .6 






64 


17.7 


8.9 


4.5 


26 


42.1 


21 .8 


11 .0 






63 


18.0 


9.1 


4.5 


25 


43 .6 


22 .6 


11 .4 


"B 





28 LABORATORY ASTRONOMY 

10 cm. cross, and on one edge for the thickness of the cross. By 
means of these graduations the angle subtended by the cross in any 
position is read directly from the scale, quarters or thirds of a 
degree being estimated and recorded in minutes of arc. 
The observations are : 

1900. 



January 2. 5 h 15 m . 




Moon to e Pegasi, 


35° 45' 


" " Altair, 


26 30 


" " Fomalhaut, 


41 40 


January 3. 6 h m . 




Moon to e Pegasi, 


23° 30' 


" " Altair, 


29 20 


" " jS Aquarii, 


8 20 


January 4. 5 h 20 m . 




Moon to e Pegasi, 


17° 40' 


" " j8 Aquarii, 


8 30 


" "5 Capricorni, 


9 45 


January 6. 5 h 50 m . 




Moon to 7 Pegasi, 


12° 0' 


" "a Pegasi, 


16 40 


" " e Pegasi, 


33 30 


January 7. 5 h 45 m . 




Moon to 7 Pegasi, 


9°40 / 


" " /3 Arietis, 


19 45 


" "a Andromedse 


, 21 15 


" " /3 Ceti, 


27 30 


January 8. 6 h m . 




Moon to a Arietis, 


11° 0' 


" "7 Pegasi, 


21 30 


January 9. 10 h m . 




Moon to a Arietis, 


9° 45' 


" " Alcyone, 


16 


" " a Ceti, 


15 30 



To represent these observations on the star map, open the com- 
passes until the distance of the pencil point from the steel point is 
equal to the measured distance — making use for this purpose of 
the scale of degrees in the margin, and then with the steel point 



THE MOON S PATH AMONG THE STARS 



29 



carefully centered on the comparison star, strike a short arc with 
the pencil point near the estimated position of the moon ; the inter- 
section of any two of these arcs fixes the position of the moon. If 
the different stars give different points, those nearest the moon may 





• 












JO - 








* 




V* 








Pkeiades 
.* 


<* 

• 


JirUUs 








20 


• 

* 

htftule-s 


. ^ 


>Jk 


X 




« 












A; 












• 








X 








® 

Mira. 


• 


• 




-iB — 










• 






-m— 



Fig. 28 



be assumed to give results nearer the truth. Fig. 28 shows the 
positions of the moon January 6 to January 9 as plotted from the 
above measures. 

Length of the Month. — If it happens that one of the positions ob- 
served in the second month falls between the places obtained on two 
successive days^ of the first month, or vice versa, a determination of 
the moon's sidereal period may be made by interpolation. Thus, on 
plotting the observation of December 12 (p. 22), which places the 
moon between the two observations on January 8 d 6 h m and Janu- 
ary 9 d 10 h m , its distance from the former is 6°.0 and from the latter 
10°. 0, while the interval is 28 h ; the moon's place on December 12 at 

12 h m is therefore the same as on January 8 at 6 h + — X 28 h , or Jan- 
uary 8 d 16\5, that is, January 9 at 4 h 30 m a.m., and the interval 
between these two times is 27 d 4 h 30 m , which is the time required for 
the moon to make a complete circuit among the stars or the length 



30 LABORATORY ASTRONOMY 

of the sidereal month. This is a fairly close approximation ; the 
observation of December 12 having been made nnder favorable 
circumstances, the configuration being well defined and the stars 
near, so that the position on that date by alignment is nearly as 
accurate as those determined by the measures on January 8 and 9. 
After three months the moon comes nearly to the same position at 
about the same time in the evening, so that it is convenient to deter- 
mine its period without interpolation by observing the time when the 
moon comes into the same star line as at the previous observation ; 
moreover, the interval being three months, an error of an hour in 
the observed interval causes an error of only 20 m in the length of 
the month. 

THE MOON'S NODE 

When a sufficiently large number of observations have been plot- 
ted to give a general idea of the moon's path among the stars, a 
smooth curve is to be drawn as nearly as possible through all the 
points and this curve should be compared with the ecliptic, as shown 
on the map. Its greatest distance from the ecliptic and the place 
where it crosses the ecliptic — the position of the node — should be 
estimated with all possible precision. For this purpose, only the 
more accurate positions obtained by the cross-staff should be used. 

After a few observations of alignment are made, the student will 
desire to use the more accurate method at once, but it is better to 
have at least one month's observation by the first method (even if 
the cross-staff is also used) for comparison with later observations by 
alignment for the purpose of determining the length of the month, 
as suggested above, without any instrumental aid whatever. 

The records of the positions of the node should be preserved by 
the teacher for comparison from year to year to show the motion 
of this point along the ecliptic. The node, as determined by the 
observations above given, was nearly at the point where the ecliptic 
crosses the line from y Orionis to Capella. Observations made in 
November, 1897, by the method of Chapter IV, gave its place on the 
ecliptic at a point where the latter intersects a line drawn through 
Castor and Pollux, thus indicating a motion of about 40° in the 
interval. 



the moon's path among the stars 31 

Observations made with the cross-staff are sufficiently accurate 
to show that the motion of the moon is not uniform, but as the dis- 
tortion of the map complicates the treatment of this subject, we 
shall defer its consideration until the method of Chapter V has 
been introduced. 

It will be well, however, as soon as measures with the cross-staff 
are begun, to devote a few minutes each evening to measures of the 
moon's diameter with an instrument measuring to 10", such as a good 
sextant ; or, better, a telescope provided with a micrometer, in order 
to show the variations of the moon's apparent size at different parts 
of its orbit. The relative distances of the moon from the earth as 
inferred from these measures should be compared with the varia- 
tions of her angular motion as read off from the chart; although 
on account of the distortion referred to above, it will not be possible 
to show more than the fact that when the moon is nearest, her 
angular motion about the earth is greatest, and vice versa. 

The sextant or micrometer may henceforward be used also for 
observations of the sun's diameter, which should be measured as 
often as once a week for a considerable period. 

When the moon's diameter is measured, a rough estimate of her 
altitude should be made in order to make the correction for aug- 
mentation in a future more accurate discussion of the measures for 
determining the eccentricity of her orbit. 

DETERMINING THE ERRORS OF THE CROSS-STAFF 

Observations with the cross-staff are most easily made just before 
the end of twilight or in full moonlight, so that the cross may be 
seen dark against a dimly lighted background. When used for 
measuring the distance of stars in full darkness, it is convenient 
to have a light so placed behind the observer that, while invisible 
to him, it shall dimly illuminate the arms of the cross. 

As the angles which are determined by the cross-staff, especially if 
large, are affected by the observer's habit of placing the eye too near 
to or too far from the end of the staff, it is a good plan to measure 
certain known distances and thus determine a set of corrections to 
be applied, if necessary, to all measures made with that instrument. 



32 



LABORATORY ASTRONOMY 



The following table gives the distances between certain stars always 
conveniently placed for observation in the United States, together 
with the results of measures made upon them with a cross-staff 
held in the hands without support, and indicates fairly the accuracy 
which may be obtained with this instrument. The back of the 
observer was toward the window of a well-lighted room, and the 
cross was plainly visible by this illumination. 



Stars 


True 
Dis- 
tance 


Measured Dis- 
tances 


Mean 


Correc- 
tion 


aUrsse Majoris to/3 Ursse Majoris 


5°. 4 


5°.8 — — 


5°.8 


-0°.4 


a ' 


t u a y u k 


10 .0 


10 .5 10°. 6 10°. 6 


10 .6 


-0 .6 


a ' 


' " " e " 


15 .2 


15 .6 15 .5 15 .7 


15 .6 


-0 .4 


P ' 


( U It ■> (( t( 


19 .9 


20 .0 20 .3 20 .2 


20 .2 


-0 .3 


a ' 


ci u v tt 


25 .7 


26 .6 26 .0 26 .1 


26 .2 


-0 .5 


a ' 


' " " Polaris 


28 .5 


29 .0 29 .2 28 .9 


29 .0 


-0 .5 


P ' 


l n u tt 


33 .9 


35 .1 34 .7 34 .4 


34 .7 


-0 .8 


V ' 


; tt tt a 


41 .2 


42 .2 42 .0 42 .0 


42 .1 


-0 .9 



The measured distances are about one-half degree too large, and 
if a correction of this amount is applied to all angles measured by 
this instrument up to 30°, the corrected values will seldom be so 
much as half a degree in error, and the mean of three readings will 
probably be correct within a quarter of a degree. 



CHAPTER III 
THE DIURNAL MOTION OF THE STARS 

As the observations of the moon require bnt a few minutes each 
evening, observations may be made on the same nights upon the 
stars. The first object is to obtain the diurnal paths of some of the 
brighter stars, and as they cast no shadow we must have recourse 
to a new method of observation to determine their positions in the 
sky at hourly intervals. 

A simple apparatus for this purpose is represented in Fig. 29. 
A paper circle is fastened to the leveling board used in the sun 




Fig. 29 



observations so that the zero of its graduation lies as nearly as 
possible in the meridian, and a pin with its head removed is placed 
upright through the center of the circle. 

A carefully squared rectangular block about 10 inches by 8 inches 
by 2 inches is placed against the pin so that the angle which its 
face makes with the meridian may be read off upon the horizontal 

33 



34 



LABORATORY ASTRONOMY 



circle. A second paper circle is attached to the face of the block 
with the zero of its graduations parallel to the lower edge ; a light 
ruler is fastened to the block by a pin through the center of its 
circle ; the ruler may be pointed at any star by moving the block 
about a vertical axis till its plane passes through the star, and then 
moving the ruler in the vertical plane till it points at the star ; a 
lantern is necessary for reading the circles and for illumination of 
the block and ruler in full darkness ; it should be so shaded that 
its direct light may not fall on the observer's eye. Sights attached 
to the ruler make the observation slightly more accurate, but also 
rather more difficult, and without them the ruler may be pointed 
within half a degree, which is about as closely as the angles can be 
determined by the circles. 



THE ALTAZIMUTH 

An inexpensive form of instrument for measuring altitude and 
azimuth is shown in Fig. 30. Here the ruler provided with 
sights A, B is movable about d, the center of the semicircle E. 

This semicircle is movable about an axis 
perpendicular to the horizontal circle F, 
and its position on that circle is read 
off by the pointer g, which reads zero 
when the plane of E is in the meridian. 
The circle F is mounted on a tripod 
provided with leveling screws. If the 
circle is so placed that the pointer reads 
zero when the sight-bar is in the mag- 
netic meridian, then its reading when 
the sights are pointed at any star will 
give the magnetic bearing of the star. 
It will, however, be more convenient to 
adjust the instrument so that the pointer 
reads zero when the sight-bar is in the 
true meridian. 

To insure the verticality of the standard a level is attached to 
the sight-bar, and by the leveling screws the instrument must be 




Fig. 30 



THE DIURNAL MOTION OF THE STARS 



35 



adjusted so that the circle E may be revolved without causing the 
level bubble to move. (See page 3G.) 

A more convenient and not very expensive instrument is the 
altazimuth or universal instrument shown in Fig. 31, which contains 
some additional parts by the use of which it may be converted into 
an equatorial instrument. 
(See page 45.) It consists 
of a horizontal plate carry- 
ing a pointer and revolving 
on an upright axis which 
passes through the center 
of a horizontal circle grad- 
uated continuously from 0° 
to 360°. The plate carries 
a frame supporting the 
axis of a graduated circle; 
this axis is perpendicular 
to the upright axis, and the 
circle is graduated from 0° 
to 90° in opposite directions. 
Attached to the 
circle is a tele- 
scope whose op- 
tical axis is in 
the plane of the 
circle. The cir- 
cle is read by 
a pointer which 
is fixed to the 

frame carrying its axis and reads 0° when the optical axis of the 
telescope is perpendicular to the upright axis. A level is attached 
to the telescope so that the bubble is in the center of its tube when 
the telescope is horizontal. In what follows, all these adjustments 
are supposed to be properly made by the maker. 




36 



LABORATORY ASTRONOMY 



ADJUSTMENT OF THE ALTAZIMUTH 

If the altazimuth is so adjusted that the upright axis is exactly 
vertical, and if we know the reading of the horizontal circle when 
the vertical circle lies in the meridian, we may determine the 
position of a heavenly body at any time by pointing the telescope 
upon it and reading the two circles. The difference between the 
reading of the horizontal circle and its meridian reading is the azi- 
muth, and the reading of the vertical circle is the altitude of the 
body. Before proceeding to the observation of stars, it will be well 
to repeat our observations on the sun, using this instrument, and 
making them in such a manner that we may at the same time get a 
very exact determination of the meridian reading by the method 
suggested on page 14. 

Place the instrument upon the table used for the sun observation ; 
bring the reading of each circle to 0° ; and turn the whole instru- 
ment in a horizontal plane until the telescope points approximately 
south, using the meridian determination obtained from the shadow 
observations. One leveling screw will then be nearly in the meridian 
of the center of the instrument, while the two others will lie 
in an east and west line. Bring the level bubble to the middle 
of its tube by turning the north leveling screw; then set the 
telescope pointing east; and "set" the level by turning the east 
and west screws in opposite directions. Be careful to turn them 
equally ; this can be done by taking one leveling screw between the 




Fig. 32 



finger and thumb of each hand, holding them firmly, and turning 
them in opposite directions by moving the elbows to or from the 
body by the same amount. Turn the telescope north, and the bubble 



THE DIURNAL MOTION OF THE STARS 



37 



should remain in place ; if it does not, adjust the north screw. The 
instrument is very easily and quickly adjusted by this method. 
The upright axis is vertical when the telescope can be turned about 
it into any position without displacing the bubble. 

Determination of the Meridian and Time of Apparent Noon. — After 
completing the adjustment of the instrument, the reading of the circle 





Fig. 33 



when the telescope is in the meridian is determined as follows : Point 
the telescope upon the sun approximately. Place a sheet of paper or 
a card behind it, and turn the telescope about the vertical axis until 
the shadow of the vertical circle is reduced to its smallest dimensions 
and appears as a broad straight line. By moving the telescope 
about the horizontal axis, bring the shadow of the tube to the form 
of a circle ; in this circle will appear a blurred disk of light. Draw 
the card about 10 inches back from the eyepiece, and pull out the 
latter nearly -J of an inch from its position when focused on distant 
objects and the disk of light becomes nearly sharp ; complete the 
focusing of this image of the sun by moving the card to or from 
the eyepiece. The distance of the card and the drawing out of the 
eyepiece should be such that the sun's image shall be about -J- to f 
of an inch in diameter. Now move the telescope until the image is 
centered in the shadow of the telescope tube, note the time, and read 
both circles ; this observation fixes the altitude and azimuth of the 



38 LABORATORY ASTRONOMY 

sun. For determining the meridian it is not necessary that the 
time should be noted, but it will be convenient to use these obser- 
vations for a repetition of the determination of the sun's path, deter- 
mining the altitudes and azimuths by this more accurate method. 

This observation should be made at least as early as 9 a.m. 
Now increase the reading of the vertical circle to the next exact 
number of degrees, and follow the sun by moving the telescope 
about the vertical axis. After a few minutes the sun will be again 
centered by this process. Note the time, and read the horizontal 
circle. Increase the reading of the vertical circle again by one 
degree to make another observation, and so on for half an hour. 
Observations may be made at one-half degree intervals of altitude, 
but those upon exact divisions will evidently be more accurate. If 
circumstances admit, observations may be made, during the period 
of two hours before and after noon, for the purpose of plotting the 
sun's path ; but, owing to the slow change of altitude in that time, 
the corresponding azimuths are not well determined, and they will 
be nearly useless for placing the instrument in the meridian. 

Some time in the afternoon, as the descending sun approaches the 
altitude last observed in the forenoon, set the vertical circle upon the 
reading corresponding to that observation, and repeat the series in 
inverse order ; that is, decrease the readings of altitude by one degree 
each time, and note the time and the reading of the horizontal circle 
when the sun is in the axis of the telescope at each successive 
altitude. 

Since equal altitudes correspond to equal azimuths (see page 14), 
east and west of the meridian, the difference of the horizontal read- 
ings is twice the azimuth at either of the two corresponding obser- 
vations (360° must be added to the western reading, if, as will 
generally be the case, the 0° point lies between the two readings). 
Therefore, one-half this difference added to the lesser or subtracted 
from the greater reading gives the meridian reading. The same 
value is more easily found by taking half the sum of the two read- 
ings. In the same way one-half the interval of time between the 
two observations added to the time of the first reading gives the 
watch time of the sun's meridian passage, or apparent noon, as it 
is called. 



THE DIURNAL MOTION OF THE STARS 



39 



Each pair of observations gives the value of the meridian reading 
and of the watch time of apparent noon ; their accordance will give 
an idea of the accuracy of the observations. 

The following observations of the sun were made March 8, 1900, 
with an instrument similar to that shown in Fig. 33. 







Time 


Altitude 


Horizontal 
Circle 




Time 


Altitude 


Horizontal 
Circle 


1 


8» 


54™ 37 s 


27°. 5 


307°.0 


9 


2 h 32 m 


10 s 


31°.0 


47°. 7 


2 


8 


58 10 


28 .0 


308 .4 


10 


2 36 


30 


30 .5 


48 .7 


3 


9 


1 42 


28 .5 


309 .2 


11 


2 39 


45 


30 .0 


49 .45 


4 


9 


4 51 


29 .0 


310 .0 


12 


2 43 


27 


29 .5 


50 .35 


5 


9 


9 5 


29 .5 


310 .9 


13 


2 47 





29 .0 


51 .15 





9 


12 20 


30 .0 


311 .8 


14 


2 50 


17 


28 .5 


51 .95 


7 


9 


15 35 


30.5 


312 .6 


15 


2 54 


7 


28 .0 


52 .85 


8 


9 


19 37 


31 .0 


313 .45 


16 


2 57 


33 


27 .5 


53 .6 



The 1st and 16th of these observations give for the meridian 
reading £ [307.6 + (53.60 + 360)] = 360°.60, and for the correspond- 
ing watch time £[8 54 37 + (2 57 33 + 12 h )] = ll h 56 m 5 s . 

Taking the corresponding a.m. and p.m. observations in this man- 
ner, we find for the eight pairs of observations above the following 
values. 

Meridian Reading Watch Time of Noon 

360°.6 ll h 56™ 5.0^ 

360 .625 56 8.5 

360 .575 55 59.5 

360 .575 55 55.5 

360 .625 56 16.0 

360 .625 56 2.5 

360 .65 56 2.5 

360 .575 55 53.5 



Altitude 

27°.5 



28 .0 

28 .5 

29 .0 

29 .5 

30 .0 

30 .5 

31 .0 
mean 



360.61 



11 56 2.< 



The agreement of these results is closer than will usually be 
obtained, the observations being made by a skilled observer and the 
angles carefully read by means of a pocket lens, which in many 
cases enabled readings to be made to 0°.05 ; any reading such as that 
of the 8th observation, where the value was estimated to lie between 
two tenths, being recorded as lying halfway between them. This 
practice adds little to the accuracy if several observations are made, 
and is not to be recommended to beginners. 



40 LABOEATORY ASTRONOMY 

MERIDIAN MARK 

It will be convenient to fix a meridian mark for future use. This 
may be done by fixing the telescope at the meridian reading, turning 
it down to the horizontal position, and placing some object (as a 
stake) at as great a distance as possible, so that it may mark the 
line of the axis of the telescope when in the meridian. A mark on 
a fence or building will serve if at a greater distance than 50 feet, 
though a still greater distance is desirable. For setting the tele- 
scope upon the mark, it is convenient to have two wires crossing in 
the center of the field of view, but the setting may be made within 
0°.l without this aid. Having established such a mark, set the 
horizontal circle at 0°, and move the whole base of the instrument 
until the telescope points upon the meridian mark. Level carefully ; 
then set the telescope again, if the operation of leveling has caused 
it to move from the meridian mark ; level again, and by repeating 
this process adjust the instrument so that it is level and that the 
telescope is in the meridian. Then press hard on the leveling 
screws, and make dents by which the instrument can be brought 
into the same position at any future time. 

After the a.m. and p.m. observations recorded above, the tele- 
scope was pointed upon a meridian mark established by observations 
made with the shadow of a pin, and the reading of the horizontal 
circle was 359°.8. The mark was then shifted about a foot toward 
the west, and the telescope again pointed upon it. As the reading 
pi the circle was then 360°.6, it may be assumed that the mark was 
now very nearly in the meridian. 

If circumstances are such that no point of reference in the meridian 
is available, it will be necessary, after determining the meridian 
readings by the sun, to set the telescope upon some well-defined object 
in or near the horizontal plane and read the circle. The difference 
between this reading and the meridian reading will be the azimuth 
of the object. Set the pointer of the horizontal circle to this value, 
and set the telescope upon the reference mark by moving the whole 
base as before. If the pointer of the circle is now brought to 0°, the 
telescope will evidently be in the meridian ; and the position is to 
be fixed by making dents with the leveling screws as before. 



THE DIURNAL MOTION OF THE STARS 41 



CHOICE OF STARS 

We are now ready to begin observations of the stars. 

The most familiar group of stars in the heavens is, no doubt, that 
part of the Great Bear which is variously called the Dipper, Charles's 
Wain, or the Plough. 

At the beginning of October, at 8 o'clock in the evening, an 
observer anywhere in the United States will see the Dipper at an 
altitude between 10° and 30° above the N.W. horizon. Set the 
telescope upon that star which is nearest the north point of the 
horizon ; read both circles to determine its altitude and azimuth, 
and note the time. Even if the telescope is provided with cross- 
hairs, the illumination of the light of the sky will not be suffi- 
cient to render them visible ; but sufficient accuracy in pointing is 
obtained by placing the star at the estimated center of the field. 
Observe in succession the altitude and azimuth of the other six 
stars forming the Dipper, noting the time in each case. 

Using the Dipper as a starting point, we will now identify and 
observe a few other stars.* The total length of the Dipper is about 
25°. Following approximately a line drawn joining the last two 
stars of the handle of the Dipper, at a distance of about 30°, we 
come to a bright star of a strong red color, much the brightest in 
that portion of the heavens ; this is Arcturus. Observe its altitude 
and azimuth, and note the time as before. Almost directly over- 
head, too high to be conveniently observed at this time, is a bril- 
liant white star, Vega (a Lyrse). A little east of south from Vega, 
at an altitude of about 60°, is a group of three stars forming a line 
about 5° in length. The central and brightest star of the three is 
Altair (a Aquilse), and its position should be observed. 

Diurnal Paths of the Stars. — Proceed in this way for about an 
hour, observing also, if time permits, the group of five stars whose 
middle is at azimuth 220° and altitude 35°. This is the constella- 
tion of Cassiopeia. Another interesting asterism will be found — 
supposing that by this time it is 9 o'clock — at azimuth 270° and 
altitude 45°, consisting of four stars of about equal magnitude, 

* Many of the latest text-books on astronomy contain small star maps which 
are valuable aids in the identification of the less conspicuous groups. 



42 LABORATORY ASTRONOMY 

placed at the corners of a quadrilateral whose sides are about 15° 
in length, and forming what is called the Square of Pegasus. 

It is convenient as an aid in identification to note in each case the 
magnitude of the star observed. As a rough standard of compari- 
son, it may be remembered that the six bright stars of the Dipper 
are of about the second magnitude; that at the junction of the 
handle and bowl is of the fourth. The three stars in Aquila are of 
the first, third, and fourth magnitudes. Yega and Arcturus are 
each larger than an average first magnitude star. The brightest 
stars in the constellation Cassiopeia and in the Square of Pegasus 
range from the second to the third magnitude. 

The little quadrilateral of fourth magnitude stars about 15° east 
of Altair and known as Delphinus, or vulgarly as Job's Coffin, may 
be observed. 

At the expiration of an hour, set again upon the Dipper stars and 
repeat the series, going through the same list in the same order. Arc- 
turus will have sunk so low in a couple of hours as to be beyond the 
reach of observation, even if the place of observation affords a clear 
view of the horizon. Yega, however, will be less difficult to observe, 
and may be now added to the list. We should not omit to make 
an observation of the pole star, which, as its name indicates, may 
be found near the pole and can be easily found, since the azimuth 
of the pole is 180°, and its altitude is equal to the latitude of the 
place. 

Prom the observed values of altitude and azimuth plot the suc- 
cessive places on the hemisphere exactly as in the case of the sun, 
and thus represent upon the hemisphere the paths of a number of 
stars in various parts of the heavens. It will be found that these 
paths are all circles of various dimensions, and that the circles are 
all parallel to the equator, as determined from the sun observations, 
that is, they have the same pole as the diurnal circles of the sun. 

At this stage it is a good plan to devote some attention to the 
representation of the various results as shown on the hemisphere, 
by means of figures on a plane surface, that is, to make careful free- 
hand drawings of the hemisphere and the circles which have now 
been drawn upon it as seen from various points of view. This is 
an important aid to the understanding of the diagrams by which it 



THE DIURNAL MOTION OF THE STARS 43 

is necessary to explain the statement and solution of astronomical 
problems ; with this purpose in view the drawings should be lettered 
and the definitions of the various points and lines written under 
them. 

ROTATION OF THE SPHERE AS A WHOLE 

So far the result of our observations is to show that the heavenly 
bodies appear to move as they would if they were all attached in 
some way to the same spherical shell surrounding the earth, and 
were carried about by a common revolution, as if the shell rotated 
on a fixed axis, passing through the point of observation. The sun 
may be conceived as carried by the same shell, but observations at 
different dates show that its place on the shell must slowly change, 
since its declination changes slightly from day to day. 

If these observations on the stars are repeated ten days or one 
hundred days later, we shall find that the declinations determined 
from them are the same ; that is, the declinations of the diurnal 
paths of the stars do not change like that of the sun. It will appear 
also that, as in the case of the sun, equal arcs of the diurnal circle 
and consequently equal hour-angles are described in equal times. 
It follows from this, of course, that stars nearer the pole will appear 
to move more slowly, since they describe paths which are shorter 
when measured in degrees of a great circle, as may be shown by 
measuring the diurnal circles on the hemisphere by a flexible milli- 
meter scale, 1 mm. being equal to 1° of a great circle on our 
hemisphere. 

If the field of view of our telescope is 5°, a star on the equinoc- 
tial will be carried across its center by the diurnal motion in 20 
minutes, while a star at a declination of 60° will remain in the field 
for twice that time, since its diurnal circle is only half as large as 
the equinoctial and an angular motion of 10° of its diurnal circle is 
only 5° of great circle. Since the declinations of the stars do not 
change, it is unnecessary to make our observations of the stars on 
the same night ; or, rather, observations made on different nights 
may be plotted as if made on the same night. We may thus obtain 
extensions of the diurnal circles by working early on one evening 
and at later hours of the night on following occasions. 



44 LABORATORY ASTRONOMY 

POSITIONS FIXED BY HOUR-ANGLE AND DECLINATION; 
THE EQUATORIAL 

It is evident that we have, in the honr-angles and declinations 
of the stars, another system of coordinates on the celestial sphere 
by means of which their position may be fixed. The altitude and 
azimuth refer the position of the star to the meridian and to the 
horizon ; while the hour-angle and declination refer its position to 
the meridian and the equator. We have hitherto found it more 
convenient to deal with the first set of coordinates, but it is often 
desirable to determine the hour-angle and declination of a body by 
direct observation, and this may be done by means of an instrument 
similar to the altazimuth but with the upright axis pointed to the 
pole of the heavens, so that the horizontal circle lies in the plane of 
the equator. With this instrument the angles read oif on the circle 
which is directly attached to the telescope measure distances along 
the hour-circle, perpendicular to the equator, i.e., declinations, while 
an angle read off on the other circle measures the angle between the 
meridian and the hour-circle of the star at which the telescope points, 
and is therefore the star's hour-angle. The two circles are there- 
fore appropriately called the declination circle and the hour-circle 
of the instrument. As these terms are used with another meaning 
as applied to circles on the celestial sphere, it would seem that there 
might be confusion from their use in this sense, but in practice it 
is never doubtful whether " circle " means the graduated circle of 
an instrument or a geometrical circle on the surface of the sphere. 

It is here supposed that the instrument has been so adjusted 
that both circles read 0° when the telescope is in the plane of the 
meridian and points at the equator. An instrument so mounted is 
called an equatorial instrument. Our altazimuth is adapted to this 
purpose by constructing the base so that it may be revolved about a 
horizontal axis perpendicular to the plane in which the altitude 
circle lies when the azimuth circle reads 0°. If, then, it has been 
placed in the meridian by the observation of equal altitudes as 
before described, it may be inclined about this latter axis through 
an angle equal to the complement of the latitude, and thus brought 
into the proper position for observing declination and hour-angle 



THE DIURNAL MOTION OF THE STARS 



45 



directly. An instrument so constructed is called a " universal " 
equatorial. To adjust the universal equatorial so that the axis 
points to the pole, adjust it as an altazimuth with both circles 
reading 0° and level it with the telescope in the meridian pointing 
south. Depress the telescope till the reading of the vertical circle 
equals the co-latitude. Tip the whole instrument so as to incline 
the vertical axis to- 
ward the north till 
the bubble plays and 
the telescope is hori- 
zontal ; to do this the 
vertical axis must 
have been tipped 
back through an 
angle equal to the co- 
latitude, and it will 
be in proper adjust- 
ment directed toward 
a point in the merid- 
ian whose altitude 
is equal to the lati- 
tude. (Fig. 34 shows 
the instrument 
adjusted for latitude 
45 s .) 

A notch should be cut in the iron arc at the bottom of the coun- 
terpoise, into which the spring-catch may slip when the adjustment 
is correct, so that the instrument may be quickly changed from one 
position to the other. If the notch is not quite correctly placed, 
the final adjustment may be made by a slight motion of the north 
leveling screw to bring the level exactly into the horizontal position, 
the vertical circle having been set to the co-latitude for this purpose. 

The proper adjustment of the altazimuth is simpler, since it 
depends only on the use of the level, while to place an equatorial 
instrument in position we must know the latitude as well. On com- 
paring the two systems of coordinates, it is clear that, while the 
altitude and azimuth both change continuously, but not uniformly 




Fig. 34 



46 LABORATORY ASTRONOMY 

with the time, the hour-angle changes uniformly with the time, 
and the declination remains the same. One advantage of the 
latter system of coordinates is that in repeating our observations 
on the same star after the lapse of an hour, we need only set 
the declination circle to the previously observed declination, and 
set the hour-circle at a reading obtained by adding to the former 
setting the elapsed time in hours reduced to degrees by multiply- 
ing by 15 ; we shall then pick up the star without difficulty. This 
is an important aid in identifying stars, which has no counterpart 
in the use of the altazimuth, and we shall henceforth use this 
method of observation in preference to the other. 



CHAPTER IV 
THE COMPLETE SPHERE OF THE HEAVENS 

The study of the motions of the sun, moon, and stars has thus 
far led to the conclusion that their courses above the plane of the 
horizon can be perfectly represented by assuming the daily rotation 
from east to west of a sphere to which they are attached, or a rota- 
tion of the earth itself from west to east about an axis lying in the 
meridian and inclined to the horizon at an angle equal to the latitude 
of the place of observation, while the sun moves slowly to and from 
the equator, and the moon, like the sun, changes its declination con- 
tinually, and has also a motion toward the east on the sphere at a 
rate of about 13° in each 24 hours. The combination of the two 
motions of the moon causes it to describe a path which will be more 
fully discussed later. We shall now begin to observe the sun, to 
see if its motion among the stars resembles that of the moon in 
having an east and west component in addition to its motion in 
declination. 

The motion of the moon can be directly referred to the stars, since 
both are visible at the same time, although the illumination of the 
dust of our atmosphere, by strong moonlight, cuts us off from 
the use of the smaller stars, which cannot be seen except when 
contrasted with a perfectly dark background. 

The illumination produced by the sun, however, is so strong that 
it completely blots out even the brightest stars, so that we cannot 
apply either of the methods that we have employed in observing 
the moon. 

We are only able to see the stars, of course, when they are above 
the plane of the horizon, but it is natural to suppose that they con- 
tinue the same course below the horizon from their points of setting 
to those of their rising. This inference is confirmed by the fact that 
some of the bright stars which set within a few degrees of the north 
point of the horizon, and which we infer complete their course below 

47 



48 LABORATORY ASTRONOMY 

the horizon, may be seen actually to do so by an observer at a point 
on the earth some degrees farther north, from which they may be 
observed throughout the whole of their courses. In the case of the 
sun, the following facts lead to the same conclusion. Immediately 
after sunset a twilight glow is seen in the west whose intensity is 
greatest at the point where the sun has just set. This glow appears to 
pass along the horizon towards the north, and its point of greatest 
intensity is observed to be directly over the position which the sun 
would occupy in the continuation of its path below the horizon, on 
the assumption that it continues to move uniformly in that path. 
In high latitudes this change of position in the twilight arch can be 
followed completely around from the point of sunset to the point of 
sunrise, the highest point being due north at midnight. It is impos- 
sible not to believe that the sun is actually there, though concealed 
from our sight by the intervening earth. (Of course, too, it is now 
generally known that in very high latitudes the sun at midsummer 
is visible throughout its diurnal course.) As the sun sinks farther, 
the light of the sky decreases, the brighter stars begin to appear, 
and it is clearly impossible to resist the conclusion that they have 
been in position during the daylight, but simply blotted out by the 
overwhelming light of the sun. 

OBSERVATIONS WITH THE EQUATORIAL 

When we have fixed the idea that the heavenly sphere revolves 
as a whole, carrying with it in a certain sense all the bodies that we 
observe, the next step is to devise some means of locating the dif- 
ferent bodies in their proper relative positions on the sphere. For 
this purpose the equatorial instrument furnishes us with an admi- 
rable means of observation. The relative position of two stars is 
completely fixed when we know the position of their parallels of 
declination and their hour-circles, since the place of each star is 
at the intersection of these two circles. 

Since an observation with the equatorial gives directly the decli- 
nation and hour-angle of a star, the method of fixing the relative 
position of two stars, A and B, is as follows : 

Point the telescope at A, and read the circles ; then set on B, and 



THE COMPLETE SPHERE OF THE HEAVENS 49 

read the circles ; then again on A, and read the circles, taking care 
that the interval between the first and second observations shall be 
as nearly as possible equal to the interval between the second and 
third. Obviously the mean of the two readings of the hour-circle 
at the pointings upon A gives the hour-angle of A at the time when 
B was observed, since the star's hour-angle changes uniformly. 
The difference between this mean and the reading of the hour- 
circle when the pointing was made upon B is, therefore, the dif- 
ference between the hour-angles of the stars at the time of that 
observation ; and this fixes the relative position of their hour-circles, 
since this difference is the arc of the equator included between 
them ; their declinations are given by the readings of the declina- 
tion circles, and thus the relative position of the two stars is 
completely known. 

As an illustration of this method, we may take the following 
example : 

With the telescope pointed at A, the readings of the hour-circle 
and declination circle were 68°. 2 and 15°.l, respectively. The 
telescope was then pointed at B, and the circles read 85°.9, 28°.l, 
and finally upon A, the readings being 69°. 1, 15°.l ; the intervals 
were nearly the same, as will usually be the case, unless there is 
some difficulty in finding the second star. Of course the first star 
can be re-found by the readings at the first observation ; indeed, 
if the intervals are plainly unequal, a repetition of the observation 
may always be made at equal intervals by setting the circles for 
each star so that no time is lost in finding. 

From the above observations we infer that when the hour-angle 
of B was 85°. 9, that of A was 68°. 65 ; and, therefore, that the hour- 
circles of the two stars cut the equator at points 17°.25 apart ; 
the hour-circle of B being to the west of that of A, so that B comes 
to the meridian earlier, or "precedes" A. 

It may be noted that the observations apparently occupied a little 
less than 4 minutes, since in the whole interval the hour-angle of A 
changed by 0°.9. 



50 



LABOKATOKY ASTRONOMY 



USE OF A CLOCK WITH THE EQUATORIAL 

If the intervals between the observations are not exactly equal, 
it will still be easy to fix the hour-angle of A at the time of the 
observation on B if the ratio of the intervals is known; if, for 
instance, the first observation of A gives an hour-angle of 25°. 3, and 
the later observation an hour-angle of 26°.3, while the intervals 
are l m between the first and second observations, and 3 m between 
the second and third, the hour-angle of A at the second observa- 
tion was obviously 25°.3 -f 0°.25. We may thus obtain by " inter- 
polation " the hour-angle of A at any known fraction of the interval. 
Plainly it is an advantage to note the time of each observation for 
this purpose, as in the following observations, which were made 
Feb. 5, 1900, for the purpose of determining the relative positions 
of the stars forming the Square of Pegasus. 



Star 


Watch Time 


Decl. Circle 


HOUR-CIRCLE 


1 7 Pegasi 


7h 14m s 


+ 15°. 2 


66°. 3 


2 a Pegasi 


15 


+ 15 .2 


83 .6 


3 /3 Pegasi 


16 15 


+ 28 .1 


84 .1 


4 a Andromedse 


17 10 


+ 29 .0 


68 .3 


5 7 Pegasi 


18 30 


+ 15 .2 


67 .6 


6 7 Pegasi 


21 30 


+ 15 .1 


(69 .2) 


7 a Andromedse 


22 30 


+ 29 .1 


69 .6 


8 /3 Pegasi 


23 30 


+ 28 .1 


85 .9 


9 a Pegasi 


24 20 


+ 15 .3 


86 .0 


10 7 Pegasi 


25 30 


+ 15 .1 


69 .1 


11 7 Pegasi 


27 30 


+ 15 .1 


69 .6 



The observations here follow each other rapidly. They were made 
by an experienced observer, and the arrangement of the stars is such 
that, after setting y Pegasi, a Pegasi is brought into the field by 
moving the telescope about the hour-axis only ; we pass to /? Pegasi 
by motion around the declination axis only, to a Andromedae by 
motion about the hour-axis, and back to y Pegasi by rotation about 
the declination axis ; so that the stars are found more quickly 
than if both axes must be altered in position at each change ; in 
observations 6 to 10 the series is observed in reversed order. 



THE COMPLETE SPHERE OF THE nEAVENS 51 

If the instrument was correctly adjusted, the declination of the 
four stars was as follows : y Pegasi + 15°.14, a Pegasi 15°.25, 
ft Pegasi 28°. 1, a Andromedae 29°.05, each being determined as the 
mean of all the observations made upon the star. 

The first advantage of the recorded times is to show that the 
reading of the hour-circle in 6 was an error, probably for 68°. 2, as 
we see by comparison with the other values of the hour-angle of 
y Pegasi, which increase uniformly about 1° in each 4 minutes. It 
will be better, however, to reject the observation entirely, as it is 
not necessary to use it for the first set of observations 1 to 5, which 
we will now discuss. 

By interpolation between 1 and 5 we find that the hour-angle of 
y Pegasi at 7 h 15 m s was § of 1°.3 greater than 66°.3, or 66°.59 ; 
at 7 h 16 m 15 s it was £ of 1°.3 greater than 66°.3, or 66°.95 ; and at 
7 h 17 m 10 s it was ^h>_ f i°.3 ie SS than G7°.6, or 67°.21. As the 
hour-angles of the other stars were observed at these times, we 
can at once find the differences of their hour-angles from that of 
y Pegasi, which are as follows : a Pegasi, 17°.01 ; /? Pegasi, 17°.15 ; 
a Andromedae, 1°.09. All the hour-angles are greater than those of 
y Pegasi, so that all the stars precede y Pegasi. By using all the 
observations we may presumably obtain more accurate results, and 
it will be well, as in all cases when a considerable number of 
observations must be dealt with, to arrange the reductions in a 
more systematic manner. 

In the table on the following page the difference of hour-angle is 
obtained by subtracting the observed hour-angle in each case from 
the hour-angle of y Pegasi, so that its value is negative, if, as in 
the results given above, the stars precede y Pegasi, and positive 
if they follow it. An observation of Venus, made on the same 
occasion, is added to the list, and an additional observation of 
a Pegasi is included ; the erroneous observation of y Pegasi at 
7 h 2l m 30 s is excluded. 

The values of the hour-angle of y Pegasi at the successive times, 
as given in column 6, are computed from the following considera- 
tions, the proof of which is left to the student. If a quantity 
changes uniformly, and its values at several different times are 
known, the mean of these values is the same as the value which 



52 



LABORATORY ASTRONOMY 



Star 


Time 


Decl. 


H.A. 


H.A. OF y PEG. 


Star follows 
y Peg. 


1 Venus 


7h i 2 m 


s 


+ 4°.0 


75°.5 


65°. 86 


- 9°. 64 


2 a Peg. 


13 





+ 15 .1 


83 .1 


66 .10 


- 17 .00 


3 t Peg. 


14 





+ 15 .2 


66 .3 


66 .35 


+ .05 


4 a Peg. 


15 





+ 15 .2 


83 .6 


66 .59 


- 17 .01 


5 p Peg. 


16 


15 


+ 28 .1 


84 .1 


66 .89 


- 17 .21 


6 a Androm. 


17 


10 


+ 29 .0 


68 .3 


67 .12 


- 1 .18 


7 7 Peg. 


18 


30 


+ 15 .2 


67 .6 


67 .44 


- .16 


8 a Androm. 


22 


30 


+ 29 .1 


69 .6 


68 .44 


- 1 .16 


9 /3 Peg. 


23 


30 


+ 28 .1 


85 .9 


68 .88 


- 17 .22 


10 a Peg. 


24 


30 


+ 15 .3 


86 .0 


68 .93 


- 17 .07 


11 7 Peg. 


25 


30 


+ 15 .1 


69 .1 


69 .17 




12 7 Peg. 


27 


30 


+ 15 .1 


69 .6 


69 .65 


+ .05 



the quantity has at the mean of the times. Using this principle, 
we find the hour-angle of y Pegasi at 7 h 21 m 22 s was 68°. 15. 

Between observations 3 and. 12 it changed 3°.3 in 13J m , or 0°.244 
per minute. Assuming this rate of change, it is easy, though labori- 
ous, to compute the hour-angle at any one of the given times ; for 
example, at 7 h 12 m s the hour-angle was 68°.15 - (9|$ times 0°.244), 
or 65°. 8 6. Labor will be saved by making a table of the values at 
the even minutes by successive additions of 0°.244, from which the 
values at the observed times are rapidly interpolated. The sixth 
column contains the number of degrees by which the hour-circle of 
the star follows that of y Pegasi. The mean values for each star 
obtained from this column are as follows. 



Star 


Decl. 


DlFF. H.A. 


7 Pegasi 

Venus 

a Pegasi 

/3 Pegasi 


+ 15°.15 
- 4 .0 
+ 15 .20 
+ 28 .10 
+ 29 .05 


0°.00 

- 9 .64 

- 17 .03 

- 17 .22 


a Androm 


- 1 .17 



The true values of the declinations of these stars as determined by 
many years of observations are for y Pegasi 14°. 63, a Pegasi 14°.67, 
/3 Pegasi 27°.55, a Andromedse 28°.53. The values from our 



THE COMPLETE SPHERE OF THE HEAVENS 



53 



■30 



OfAndrom. 



tZO 



rega,?9JLs 



Venus 



observations are 15°.15, 15°.20, 28°.10, 29°.05, so that the latter 
require corrections of — 0°.52, — 0°.53, — 0°.55, and — 0°.52, respec- 
tively. This is due to a faulty adjustment of the instrument, but 
the error from this cause evidently affects all the observations by 
nearly the same amount, 0°.53, so that the relative positions are 
given quite accurately ; our observations placing the whole constel- 
lation about \° too far north. 

Since Venus is in the near 
neighborhood of y Pegasi, we 
may assume that the observa- 
tions of that planet are subject 
to the same corrections, that 
she preceded y Pegasi by 9°.64, 
and that her true declination 
was _ 4°.o - 0°.53, or - 4°.53. 
The correction is applied alge- 
braically with the same sign as 
to the other stars, since it must 
be so applied as to make the 
true place farther south than 
the observed place. 

The places of the Square of Pegasus and the planet Venus, as 
seen in the sky Feb. 5, 1900, are shown in Fig. 35. 

Before plotting the stars on the hemisphere from the above 
data, it must be prepared by drawing upon it in their proper posi- 
tions the meridian, zenith, pole, and equator. Draw the hour-circle 
of y Pegasi (see Fig. 19, p. 17) at the proper hour-angle from the 
meridian, to give its position at the time of the last observation, 
which may be determined by making it intersect the equator at the 
proper point 69°. 6 west of the meridian, and place the star upon it 
at a distance from the equator equal to the observed declination, 
15°. 14. The hour-angle of a Pegasi should be drawn in the same 
manner to cut the equator at 86°.66 from the meridian, and the 
star placed upon it at the observed declination, 15°. 20. Of course 
on the scale of so small a hemisphere the nearest half degree is 
sufficiently accurate. Remember that the configurations on the 
hemisphere and on the map are semi-inverted. 



-10 



-15 

Fig. 35 



-30 



54 LABORATORY ASTRONOMY 

CLOCK REGULATED TO SHOW THE HOUR-ANGLE OF THE 
FUNDAMENTAL STAR 

The method of calculating the hour-angles of y Pegasi in the last 
example shows that if the reading of the watch can be relied upon, 
the observations of that star need only be made at the beginning 
and at the end of the period of observation, the hour-angle at any- 
time being determined by its uniform increase ; or even from a 
single observation at the beginning of the period, since at the time 
of observation of any star the hour-angle of y Pegasi can be 
inferred from that at its first observation by adding the number of 
degrees which it would have described in the time elapsed, obtained 
by multiplying the number of hours by 15, or, what gives the same 
results, dividing the minutes by 4. Moreover, if the rate of the 
watch is such that it completes its 24 hours in the time in which 
the stars complete their daily revolution, and if its hands are so set 
as to read 12 hours when y Pegasi is on the meridian, the difference 
of hour-angle at any time will be equal to the reading taken directly 
from the hands of the watch reduced as above to degrees, for when 
the star is on the meridian and its hour-angle therefore zero, the 
watch marks h m s . Four minutes later by the watch the hour- 
angle of the star has increased by the diurnal revolution to 1°; 
in four minutes more to 2° ; when the watch indicates 1 hour, the 
star's hour-angle has increased to 15°, and so on, till 24 hours have 
elapsed, when the star will again be on the meridian and the cycle 
recommences. 

The rate of an ordinary watch is sufficiently near to that of the 
stars to allow of its use for this purpose for periods of an hour 
without causing any error in our observations. 

In the use of this method we may regard the observation of the 
fundamental or zero star as a means of finding out whether the 
clock is set to the right time : thus, in the following set of obser- 
vations the first observation gives the hour-angle of y Pegasi 67°. 6 
at 7 h 15 m 10 s , but as 67°.6 equals 4 h 30 m 24 s , we may regard the 
clock as 2 h 44 m 46 s fast ; and by applying this correction to all the 
observed times, may write down at once under the title " corrected 
time " what the readings would have been if the clock had been set 



THE COMPLETE SPHERE OF THE HEAVENS 



55 



to show hours, when the hour-angle of y Pegasi was 0°. Dividing 
these by 15 we have the hour-angle in degrees given in column 4. 

The following observations were undertaken for determining the 
configuration of the stars in Orion and its neighborhood, Feb. 6, 1900. 



Star 


Obs. T 


[ME 


Corrected 
Time 


H.A. OF 
y Peg. 


Observed 
H.A. of Star 


Decl. 


Follows 
y Peg. 


7 Pegasi 


7 h 15™ 


108 


4 h 30 m 24 s 


67°. 6 


67°. G 


+ 15°. 5 




a 


7 20 





4 35 14 


68 .8 


348 .5 


- 1 .4 


80°. 3 


b 


7 22 





4 37 14 


09 .3 


349 .95 


- .0 


79 .35 


c 


7 23 


30 


4 38 44 


G9 .7 


348 .1 


- 2 .2 


81 .8 


d 


7 25 


20 


4 40 34 


70 .1 


344 .8 


+ 7 .1 


85 .3 


e 


7 27 


10 


4 42 24 


70 .0 


353 .0 


+ 6.1 


77 .6 


f 


7 28 


45 


4 43 59 


71 .0 


347 .5 


- 9 .9 


83.5 


g 


7 30 


20 


4 45 34 


71 .4 


35G .4 


- 8 .2 


75 .0 


h 


7 32 





4 47 14 


71 .8 


351 .G 


- 5 .5 


80 .2 


i 


7 34 





4 47 14 


72 .3 


334 .7 


-IG .9 


97 .6 


J 


7 35 


45 


4 50 59 


72 .7 


321 .3 


+ 5 .05 


111 .4 


a 


7 37 


45 


4 52 59 


73.2 


352 .9 


- 1 .4 


80.3 


7 Pegasi 


7 39 


50 


4 55 4 


73 .8 


73 .9 


+ 15 .4 




Moon 


7 42 





4 57 14 


74 .3 


27 .6 


+ 20 .4 


46.7 



The results of columns 6 and 7 enable us to map the constellation 
as in Fig. 36. 

One or two constellations may be plotted in this manner both on 
the map, which shows the constellation as seen in the sky, and on 



+20° 
+10° 
0° 
-10° 
-20° 











qMooti 




















jTegasi 


Pre 


cyon 


• 


Orion 














• 














• Sir 


ins 













105° 



90° 



75' 



60° 
Fig. 36. 



45" 



30' 



15° 



56 LABORATOKY ASTRONOMY 

the hemisphere, where it is semi-inverted. It will be advisable, 
however, before much work has been done in this way, to introduce 
a slight modification. 

THE VERNAL EQUINOX— RIGHT ASCENSION 

The precession of the equinoxes causes a change in the position 
of the equator, which slowly changes the declinations of all the 
stars. For this reason it is found more convenient to select, instead 
of y Pegasi as a zero star, the point upon the equator at which the 
sun crosses it from south to north about March 21 of each year. 
This point, which is called the vernal equinox, is not fixed, but 
its motion, due to precession, is simpler than that of any star which 
might be selected as a zero point; it precedes the hour-circle of 
y Pegasi at present by about 8 minutes of time, or 2° of arc, and it 
was because of this proximity that we first selected that star. 

Instead, therefore, of adjusting our clock so that it reads h m s 
when y Pegasi is on the meridian, we set it to that time when the 
vernal equinox is in that plane ; its readings then give the hour-angle 
of the vernal equinox, and the difference between the hour-angles of 
that point and of the star may be directly obtained from our obser- 
vations. The distance by which a star follows the vernal equinox 
is called its right ascension ; more carefully defined, it is the arc 
of the equator intercepted between the hour-circle of the star and 
the hour-circle of the vernal equinox (which measures the wedge 
angle between the planes of these circles) j it is also the angle between 
the tangents drawn to these two circles where they intersect at the 
pole. Since any star which is east of the vernal equinox follows it, 
the right ascensions of different stars increase toward the east, that 
is, toward the left in the sky as we face south, but toward the right 
on the solid hemisphere as we look down from the outside upon its 
southern face. 

Hereafter we shall fix the positions of the stars by their right 
ascensions and declinations. We may make use of the observations 
already reduced with very little additional labor. Since y Pegasi 
follows the vernal equinox by 2°, we need only add that amount to 
the quantities given in column 7 on page 55 to know the right 



THE COMPLETE SPHERE OF THE HEAVENS 57 

ascension of the different stars. If we learn later that on Febru- 
ary 6 the right ascension of y Pegasi was more exactly h 8 m 5 S .64, 
we may further correct by adding 5 s , or even 5 8 .64, if the accuracy 
of the observations warrants it. The method of determining the 
exact position of the zero star with reference to the vernal equinox 
is given in Chapter VI. 

Formerly right ascensions were measured altogether in degrees, 
but owing to the modern use of clocks, it has long been customary 
to give them in hours ; for this reason the hour-circle of instruments 
mounted as equatorials is graduated to read hours and minutes 
directly. Since our universal equatorial is intended to serve also 
as an altazimuth, its circles are both graduated to degrees. 

SIDEREAL TIME 

In the last section right ascension has been defined as the angle 
between the hour-circle passing through a star and the great circle 
passing through the pole and the vernal equinox. The latter circle 
is called the equinoctial colure. We have also suggested the use 
of a clock set to read h m s at the time when the vernal equinox 
is on the meridian ; so that the hour-angle of the vernal equinox at 
any time will be given directly by the reading of the face of the 
watch in hours, minutes, and seconds, from which the angle in 
degrees is computed by dividing by 15. A clock set in this manner, 
and running at such a rate that it completes 24 hours in the time 
that the star completes its revolution from any given hour-angle to 
the same hour-angle again, is said to keep sidereal time. We 
shall find later that a clock so regulated gains about 4 minutes a 
day on a clock keeping mean time, thus gaining 24 hours on an 
ordinary clock in the course of a year, and agreeing evidently with 
a clock keeping apparent time, as defined on page 19, at that time 
when the sun is at the vernal equinox and crosses the meridian at 
the same time with the latter. 

Let us suppose now that the vernal equinox has passed the 
meridian by one hour, then its hour-angle is l h , or 15° ; and our 
sidereal clock indicates exactly l h m s . Any star which is at this 
time on the meridian, that is, whose hour-angle is 0°, must therefore 



58 



LABOKATOKY ASTRONOMY 



follow the vernal equinox by l h , or 15°, while at the same instant 
the time by our sidereal clock is l h m s . By our definition of 
right ascension, since the star follows the vernal equinox by l h , its 
right ascension is l h ; in this case, therefore, the right ascension of 
the star in hours, minutes, and seconds has the same value as the 
time given by the hands of the clock. In the same way, if the 
vernal equinox has passed the meridian so far that its hour-angle is 
2 h 15 m , the face of the clock will show 2 h 15 m ; and any star then 
upon the meridian follows the vernal equinox by 2 h 15 m . The same 
relation holds here ; namely, that the right ascension of the star is 
equal to the time by the sidereal clock when the star is upon the 
meridian. This might have been given as a definition of the term 
" right ascension " ; and, indeed, so closely are the two connected 
in the mind of the practical astronomer that if the right ascension 
of a star is given, he at once thinks of this number as representing 
the time of its meridian passage. 



RIGHT ASCENSION PLUS HOUR-ANGLE EQUALS 
SIDEREAL TIME 

We may here give an explanation of a general principle of very 
frequent application, and of which this is simply a particular case. 
Suppose the vernal equinox, represented by the symbol °f (Fig. 37), to 

have passed the meridian by 5 h 10 m . 
Then a star, S, whose right ascen- 
sion is 2 h 15 m , since it follows the 
vernal equinox by that amount, 
will have passed the meridian by 
2 h 55 m ; and its hour-angle will be 
2 h 55 m . The arc of the equator 
between the meridian and the ver- 
nal equinox may be considered as 
made up of two parts: the right 
ascension of the star, which is measured by the arc eastward 
from the vernal equinox to the hour-circle of the star, and the 
hour-angle of the star, which extends from the meridian westward 
to the hour-circle of the star. Since this is true of any star, or, 




fig. 37 



THE COMPLETE SPHERE OF THE HEAVENS 59 

indeed, of any heavenly body, we may make the following general 
statement : The right ascension of any body plus its hour-angle at 
any instant will be equal to the sidereal time at that instant ; or, 
as it is sometimes written: 11. A. + H.A. = Sidereal Time. If the 
body is a point on the meridian, its H.A. = zero ; hence the K.A. 
of a star on the meridian, or briefly, K.A. of the meridian = Sidereal 
Time, as we have before shown. 

From this relation we may most simply determine the right 
ascension of any heavenly body by observing its hour-angle with 
the equatorial instrument, and at the same time noting the sidereal 
time, since E.A. = Sidereal Time — H.A. It is by this method 
that we shall now proceed to make a somewhat extended catalogue 
of stars from which we may plot their positions upon the globe. 

We will here notice some of the important uses to which this 
principle may be put. If by any other means the right ascension 
of a body is known, we may find its hour-angle at any given sidereal 
time by the equation, Sidereal Time — K.A. = H.A. This gives us 
an easy way to point upon any object whose right ascension and 
declination are known, if we have a clock keeping sidereal time ; 
and this is the usual way in which the astronomer finds the objects 
which he wishes to observe, since they are generally so faint that 
they cannot be seen by the naked eye. For example, to point the tele- 
scope at the great nebula in Orion, whose right ascension is 5 h 28 m , 
and declination 6° S., we first set the declination circle to — 6°, 
and if the sidereal time is 7 h 30 m we set the hour-circle to 2 h 2 m , 
then the telescope will be pointed upon the star. If the sidereal 
time is 4 h 30 m , in which case the star evidently has not reached 
the meridian by nearly an hour, we must add 24 hours to the sidereal 
time ; then the expression, H.A. = Sidereal Time — E.A. will 
become H.A. = 28 h 30 m - 5 h 28 m , or 23 h 2 m , the hour-angle being 
reckoned, as before stated, from h up to 24 h . If then the hour- 
circle is brought to the reading 345 J° = 15° x 23^, we shall find 
the star in the field. 



60 LABORATORY ASTRONOMY 

THE CLOCK CORRECTION 

The same principle enables us to set our clock correctly to sidereal 
time by observing the hour-angle of any star whose right ascension 
is known. For example, the right ascension of Sirius being 6 h 40 m , 
or 100°, and its hour-angle being observed to be 330°, or 22 h , the 
sidereal time is R.A. + H.A., that is, 430°, or, subtracting 360°, is 
70°, corresponding to 4 h 40 m ; and a clock may be set to agree ; or, 
by subtracting the time which it then indicates, we determine a 
correction to be applied to its reading to give the true sidereal 
time. If, for instance, at the observation above, the clock time is 
4 h 41 m 10 s , the clock correction is — l m 10 s . In this case the clock 
is l m 10 s fast, the time which it indicates is greater than the true 
time, and its " error " is said to be + l m 10 s . On the other hand, 
when the clock is slow the correction to true time is positive, while 
the " error " is negative. 

It is customary to observe this distinction between the terms 
" error " and " correction " ; the former is the amount by which the 
observed value of a quantity exceeds its true value, while the correc- 
tion is the quantity which must be added to the observed to obtain the 
true value. They are thus numerically equal but of opposite sign. 

The error of the declination circle determined by the observations 
of page 53 was + 0°.53, while the correction was — 0°.53. 

For the constantly occurring " clock correction," we shall use the 
symbol At, the value ot which is positive if the clock is slow and 
negative if it is fast. 

If, as is often desirable, we wish to observe a body of known right 
ascension upon the meridian, we have only to observe it when the 
' time by the sidereal clock is equal to its right ascension. 

We may of course find the right ascension of the moon by a direct 
comparison with the neighboring stars, just as we have determined 
the difference in right ascension of a Pegasi, from that of y Pegasi, 
for the brighter stars can be easily observed at the same time as the 
moon ; but no star is so bright that it can be readily observed by 
our small instrument when the sun is above the horizon,* and we 
have therefore no means of making a direct comparison between 

* See, however, page 70. 



THE COMPLETE SPHERE OF THE HEAVENS 61 

a star and the sun. But by means of our clock and our new method 
of observation this becomes easy ; and the sun is to be added to the 
list of bodies whose right ascension we are to observe regularly. It 
is only necessary that we should be provided with a clock which 
keeps correct sidereal time. (See page 67.) 

We have already spoken of the means of setting the clock ; 
now a few words as to how the regularity of its rate may be deter- 
mined. It is only necessary to observe the watch time at which 
any star is at a given hour-angle on successive nights. If the 
rate of the clock is such that the interval between the observa- 
tions is greater than 24 hours, the watch is gaining ; if the amount 
is less than half a minute a day, the watch may be assumed for our 
purposes to be keeping correct sidereal time, its actual error at any 
time being checked, as before described, by the observation of the 
hour-angle of bodies of known right ascension. 

LIST OF STARS 

Our first care will be to observe a number of bright stars not very 
far from the equator which will serve for setting the clock or deter- 
mining its error, selecting them so that several shall always be above 
the horizon and may at any time be used for this purpose. Several 
of those already observed will be found in the list given on the 
following page, which contains the approximate places of a number 
of conspicuous stars. 

By repeated comparisons of these stars with each other and with 
y Pegasi, their right ascensions may easily be fixed within 30 s , and 
they may then be used for determining the clock error at any time 
when they are visible. The observations of each evening should be 
reduced as soon as possible and maps made of the various constella- 
tions similar to those of Figs. 35 and 36 ; it is, however, impossible 
to represent any large portion of the sphere satisfactorily on a 
plane surface, and, in order to have a proper idea of the relative 
positions of the various constellations, we must plot our results on 
a globe — a proceeding still more necessary when we come to study 
the motion of the sun and moon among the stars by the method of 
the following chapter. 



62 



LABORATOKY ASTRONOMY 



A globe 6 inches in diameter is sufficiently large for our purpose ; 
it should be so mounted that it may be turned about its axis on a 
firm support, and upon it should be traced 24 hour-circles 15° apart, 
and small circles (parallels of declination) parallel to the equator 
and 10° apart ; its surface should be smooth and white, and of such 
a texture as to take a lead-pencil mark easily, but permit of erasure. 

TIME STARS 



Star 


Mag. 


K.A. 


8 


Stab 


Mag. 


R.A. 


S 


7 Pegasi 


3 


h .l 


+ 15° 


Denebola 


2 


llh.7 


+ 15° 


j8 Ceti 


2 


.6 


- 19 


5 2 Corvi 


3 


12 .4 


- 16 


/3 Andromedse 


2 


1 .1 


+ 35 


Spica 


1 


13 .3 


- 11 


a Arietis 


2 


2 .0 


+ 23 


Arcturus 


1 


14 .2 


+ 20 


a Ceti 


2* 


3 .0 


+ 4 


a 2 Librae 


3 


14 .8 


+ 16 


Alcyone 


3 


3 .7 


+ 24 


a Serpentis 


3 


15 .7 


+ 7 


Aldebaran 




4 .5 


+ 16 


Antares 


1 


16 .4 


-26 


Capella 




5 .2 


+ 45 


a Ophiuchi 


2 


17 .5 


+ 13 


Rigel 




5 .2 


- 8 


Y 2 Sagittarii 


3 


18 .0 


-30 


e Orionis 




5 .5 


- 1 


Vega 


1 


18 .6 


+ 39 


Betelgeuze 




5 .8 


+ 7 


Altai r 


1 


19 .8 


+ 9 


Sirius 




6 .7 


- 17 


a 2 Capricorni 


4 


20 .2 


- 13 


Castor 




7 .5 


+ 32 


a Delphini 


4 


20 .6 


+ 16 


Procyon 




7 .6 


+ 5 


e Pegasi 


2i 


21 .7 


+ 9 


Pollux 




7 .7 


+ 28 


a Aquarii 


3 


22 .0 


- 1 


a Hydrse 


2 


9 .4 


- 8 


a Pegasi 


2i 


23 .0 


+ 15 


Regulus 


1 


10 .1 


+ 12 











The number attached to the Greek letter indicates that the star to be 
observed is the following of two neighboring stars. 



CHAPTER V 



MOTION OF THE MOON AND SUN AMONG THE STARS 



For plotting the stars on the globe in their proper places, as given 
by their right ascensions and declinations, it is convenient to have 
the equator graduated into spaces of 10 ra each ; this may be done 
by laying the edge of a piece of paper along the equator, and mark- 
ing off the points of intersection of the equator with two consecu- 
tive hour-circles; laying the paper upon a flat surface, bisect the 
space between the two lines with the dividers, and trisect each of 
these spaces by trial, testing the equality of the spacing by the 
dividers ; this may be satis- 
factorily done by two or three 
trials, and the short scale thus 
obtained may be easily trans- 
ferred to the arcs on the equator 
between each two hour-circles. 
It may be found convenient to 
bisect each of the spaces on the 
scale, thus dividing the equator 
into spaces of 5 m each. 

A strip of parchment or 
parchment paper about 8 inches 
long and £ inch wide, of the 
shape shown in Fig. 38, and 
graduated to degrees, completes 
the apparatus necessary for 
plotting. The hole being 
placed over the axis of the 
globe, the graduated edge of 
the strip may be made to coincide with the hour-circle of any star 
by causing it to intersect the equator at a point corresponding to the 
star's right ascension, taking care that the edge lies in a great circle 

03 




£V 



Fig. 38 



64 LABORATOKY ASTRONOMY 

of the sphere ; the graduated edge gives at once the proper declina- 
tion for plotting the star upon its hour-circle, and the point may be 
marked with a well-sharpened, hard lead pencil ; the latter should 
be carefully kept, and used for purposes of plotting only. With 
this simple apparatus the stars may be rapidly and accurately 
placed upon the globe. 

An attempt should be made to represent the magnitudes of the 
stars by the size of the dots which indicate their places. 

THE MOON'S PATH ON THE SPHERE 

The moon should be placed on the list of objects for regular 
observation, the observations being made in precisely the same 
manner as those of the stars, and its place should be plotted 
upon the globe at each observation and marked by a number, 
giving the date of the month. This method of fixing the moon's 
place is much more accurate than those made use of in Chapter II, 
and, as the places are plotted upon a globe, we may study to better 
advantage those peculiarities of her motion which are masked by 
the distortion of the map referred to in Chapter II. 

The position of the node may now be fixed with such a degree of 
accuracy that its regression is shown by the observations of two or 
three months, if some care is taken to observe as nearly as possible at 
the same altitude in the successive months, so that the corrections 
for parallax may be nearly the same ; indeed, a very few months will 
force upon the notice of the observer the fact that the moon's path 
does not lie in one plane, just as observations a few days apart 
show that the sun's diurnal path is not really a small circle lying 
in one plane. 

We also study the variable motion of the moon by applying 
dividers between the successive plotted places and then placing 
the dividers against the parchment scale to measure the distance 
in degrees traversed in the plane of the orbit. The scale must 
lie along an hour-circle so as to conform to the curvature of the 
sphere. 

The average rate being about 13° a day, the points on the orbit 
should be determined as nearly as possible at which the motion is 



MOTION OF THE MOON AND SUN AMONG THE STARS 65 



greater and less than this amount, and the point of most rapid 
motion fixed as closely as possible ; this point is most simply fixed 
by its distance in degrees from the ascending node of the moon's 
orbit. Since the latter point, however, is continually changing, 
it is customary to reckon the so-called " longitude in the orbit " of 
the point by measuring from the vernal equinox along the ecliptic 
to the node, and adding the angle measured along the orbit from 
the node to the point. 

The variations of the moon's angular diameter and the point of 
the orbit where the diameter is greatest should be compared with 

w vn vr v 



w 



1 1 1 1 

-L | ' ' / 


f 1 ~ r T TT 

' 1 ' ' 1 1 


i tttt- 
I'iii 

J ! 1 I ! 


H'ITT 
| i I i i 


1 1 II 1 


1 \ \ \ \ 


Y^ — <""< — < 
\ t -^ "^" 

V V \ \ \ 


-vvy\ 


WW 




' 1 1 

I i ! ' 

i ' 1 ' 


I ! I I I 

i'Ii 
i'Ii 


Mil 1 

I'll 1 
III 1 




I • \ \ \ \ 

A- A -v * -> 

l * ' i ' 






\ „V "\" \ 


\ \ \ \ 


V \ \ v' 


' 1~ir 

i ill 

i ili! 


i-i. / / / 
/ W--H 

III! 

1: 1 ' I ' 


1 i j ! 

■-J-i-i+40_ 


1 ■■) i | 
I '•• 1 
-4-l-vi-T- 

1 '. ' V 
•l ■■ 1 1 '•■' 


i 1 M ' * 


-\"\' \ V 


\ ? y \\ 






V'v\'\ 


~r~t~-i-j~ 
1 / / < 4 


11 ' ' 
1 1 1 ' 


i-Li+3?. 

ill' 


1 1 •' ' 


\ 1 ' ^ ' 
' 1 \*ft- 


r \\ \ 




V l \ * v 


\ vt5 




i I '■' 

/ ' / ■! 


1 ' J 1 
1 j ' | 1 


! j '•••■ 


4rH 


■•■'■■■ i * i"t 

-j — l--r f 






V 1- ■* " * 


•-\ \ \ > 

^ \ \ \ 


Ik.-** 1 V 
^ \ \ \ 


Ijfj! 




-|-Uii 




! 1 1 \ 


1 I 1 \ 


\\\\ 


\ \ \» v 






I ! ' 
/ / 


LA } \_ 


{ I i 


I > ' 


11_Ul 


/\\\ 


\\\'' 


.\AA" 


v# 


i'-H'A ' v " 


' j \ \ 


i i ■' 


i !"1' 


1 ' 1 


nv 


~1 ' ' 


-L .1 1 


— j-«l--l- 


! *1- 

-r-T-1 

1 1 L 


V-v-\-* 


■ ^ \ * 


i \ \ 


\ Vv- 


\ \ \ 



+30 



10 



i-i -20 



-50 



120 



110 



100 90 



70 60 SO 40 30 20 

Fig. 39 



the results obtained from the investigation of the angular velocity 
in the orbit, since we thus gain some knowledge of the moon's 
relative distances from us at different points of its orbit, and of the 
relation between its distance and its rate of motion about the earth. 
The scale of the 6-inch globe is too small to do justice to the 
accuracy of our observations, which are accurate to a quarter or a 
tenth of a degree, and it will be interesting to plot these observations 



66 LABORATORY ASTRONOMY 

on a map constructed on a larger scale, and on a plan which 
reduces the distortion to very small limits in the region of the 
ecliptic ; such a map is shown in Fig. 39 on a reduced scale j the 
ecliptic is here taken as a straight horizontal line, as the equator 
is in the star map previously used ; the latitude, or angular distance 
of a point from the ecliptic measured on a great circle perpendicular 
to the latter, serves as the coordinate corresponding to the declina- 
tion on our former map, while right ascension is replaced by lon- 
gitude, or distance along the ecliptic measured from the vernal 
equinox up to 360°. The same map will serve also for plotting 
the paths of the planets in our later study. 

For convenience in plotting, the parallels of declination and the 
hour-circles are printed in broken lines upon the map. The obser- 
vations of the moon shown in the figure are those of December, 
1899, already plotted on the map of Fig. 25. 

THE SUN'S PLACE AMONG THE STARS 

By means of the equatorial we may also determine the place of 
the sun among the stars, although the method of direct comparison 
with stars we have used in the case of the moon is not applicable, 
since the stars are not visible when the sun is above the horizon ; 
the most obvious method which is capable of any degree of accuracy 
involves the use of a clock regulated to sidereal time. 

To determine the place of the sun, point upon it with the equa- 
torial about two hours before sunset ; note the time, and read the 
circles ; as soon as possible after sunset observe a star in the same 
manner, with the instrument as near as may be to its position at 
the sun observation. It is evident that if the circumstances were 
fortunately such that the telescope did not have to be moved between 
the observations, the difference in right ascension of the sun and the 
star would be the difference in time noted by the sidereal clock, 
while the declinations of the sun and star would be the same. The 
nearer the star is to the position in which the sun was observed, 
the less will be the errors arising from imperfect adjustment and 
orientation of the instrument; while the shorter the interval be- 
tween the observations, the smaller will be the error due to the 



MOTION OF THE MOON AND SUN AMONG THE STARS 67 

uncertainty in the rate of the clock. As the condition of not 
moving the telescope can seldom be fulfilled, however, we must 
treat the observation as follows : 

Let R.A., H.A., t, and At be the right ascension, hour-angle, 
clock time, and clock correction at the time of the star observation, 
and R.A.', H.A.', t', and At, the corresponding quantities at the 
sun observation. The equation 

R.A. + H.A. = Sidereal Time = t + At 

determines the value of At, which substituted in the equation 

Sidereal Time = t' + At = R.A.' + H.A.' 

determines the value of R.A.', the sun's right ascension at the 
moment of observation. 

The value of At, as determined from the first equation, will be 
negative if the clock is fast, and positive if the clock is slow ; and 
it must always be applied to the observed time with the proper 
sign. The declination of the sun is, of course, given directly by 
the reading of the declination circle. 

The following example illustrates the method : 

March 30, 1899, an observation of the sun with an equatorial 
telescope, and a clock keeping sidereal time, gave the following 
values : 

Observed time = 5 h 36 ni 26 s ; H.A. = 75°.7 = 5 h 2 m ^8 S ; 8 = + 4°.l. 
About an hour after sunset an observation of a Ceti was made 
in nearly the same position of the instrument, which gave the 
following values : 

Observed time = 7 h 53 m 43 s ; H.A. = 74°.l = 4 h 56 m 24 s ; 8 = + 4°.2. 
This latter gives, from the known right ascension of a Ceti, 

2 h 57 m s + 4 h 56 m 24* - Sidereal Time == 7 h .53 m 43 s + At, 

and hence At = — 19 s ; and, applying the same equation to the sun 
observation, 

Sun's R.A. + 5 h 2 m 48 s = 5 h 36 m 26 s - 19 s = 5 h 36 m 7 8 ; 

hence the sun's right ascension at the time of the first observation 
was h 33 m 19 s . This is liable to an error equal to the uncertainty of 
the circle readings, which may be at least one-twentieth of a degree, 



68 LABOKATORY ASTRONOMY 

or 12 s of time, and to ail error equal to the uncertainty of the gain 
or loss of the clock during the interval of 2-j- hours between the 
two observations, probably five or ten seconds of time. We may 
assume that the errors arising from defective adjustment of the 
instrument were the same for both objects, and may be neglected, 
since the position of the instrument was very nearly the same for 
both observations. 

DIFFERENTIAL OBSERVATIONS 

The declination of a Ceti, as read from the circles, was + 4°.2, 
while its known declination was + 3°. 7. The correction necessary 
to reduce the circle reading to the true value is, therefore, — 0°.5, 
and, applying this quantity to the reading on the sun, we have for 
the true value of the sun's declination + 4°.l — 0°.5 = + 3°. 6. It is 
worthy of note that the correction is about the same as that deter- 
mined from the observations discussed on page 53, which were 
made with the same instrument in nearly the same adjustment, but 
from a different place of observation. These results indicate an 
inherent defect in the instrument, which is at least in great part 
neutralized by the method of observation. It is a very important 
thing, even with the most delicate instruments, to avail ourselves of 
methods which accomplish this object, and surprisingly good work 
may be done with poor instruments by paying proper attention 
to the details of observation for this purpose. 

Methods by which an unknown body is thus compared with a 
known body under circumstances as nearly alike as possible are 
called "differential methods." 

INDIRECT COMPARISON OF THE SUN WITH STARS 

It is often possible to determine the difference of right ascension 
of the sun and some well-known star by using the moon as an inter- 
mediary, determining the difference of right ascension of the sun and 
moon during the daytime and comparing the moon and a star as soon 
as possible after sunset, the motion of the moon during the interval 
being allowed for. The irregularity of the moon's motion may, 



MOTION OF THE MOON AND SUN AMONG THE STARS 69 

however, introduce a greater error than that arising from uncertainty 
in the rate of the clock. A better method is offered on those not 
infrequent occasions when the planet Venus is at its greatest 
brilliancy, when it may be easily observed in full daylight ; the 
motion of Venus in the interval is much smaller and more nearly 
uniform, and, therefore, more accurately determined ; and by this 
method the interval between the observations connecting the sun 
with Venus and Venus with the star may be reduced to a very few 
minutes, or even seconds, so that the error due to the clock may be 
regarded as negligible. 

The following observations illustrate the method. 



1900 



April 19.3. Procyon 
Venus . 
Procyou 
Venus . 
Procyon 

April 20.0. Sun . . 

Venus . 

Sun . . 

Venus . 

Sun . . 

April 20.3. Procyou 
Venus . 
Procyon 
Venus . 
Procyon 



Watch Time 



gh 17m 45s 

8 19 33 

8 21 45 

8 23 

8 24 53 

1 28 45 

1 31 35 

1 3G 10 

1 38 33 

1 41 21 

9 31 27 
9 32 30 
9 33 28 
9 35 
9 36 



H.A. 



15°. 4 

56 .1 

16 .4 

57 .0 

17 .2 

358 .5 
313 .2 

.15 
315 .0 

1 .4 

33 .25 

72 .9 

33 .9 

73 .3 

34 .25 



+ 5°. 5 
+ 26 .1 
+ 5 .55 
+ 26 .05 
+ 5 .30 

+ 11 .6 
+ 25 .35 

+ 11 .6 
+ 25 .3 
+ 11 .6 

+ 5 .45 

+ 25 .9 
+ 5 .4 
+ 25 .9 
+ 5.4 



The observations April 19.3, that is, April 19 about 7 p.m., give for 
the hour-angle of Venus 5 6°. 55 at the watch time 8 h 21 m 17 s , and for 
that of Procyon 16°.33 at 8 h 21 m 28 s ; hence at 8 h 21 m Procyon 
followed Venus 40°.22. 

In the same way we find that April 20.3 Procyon followed Venus 
39°.3, the change of the right ascension of Venus being 0°.92 in 25.2 
hours. A simple interpolation shows that April 20.0 Procyon 



70 LABORATORY ASTRONOMY 

followed Venus 39°.59, and the observations at that time show that 
Venus followed the sun 45°. 92, so that Procyon followed the sun 
39°.59 -f- 45°. 92 = 85°.51, and the difference of right ascension between 
Procyon and the sun at noon on April 20 was, therefore, 5 h 42 m 2 8 . 

ADVANTAGES OF THE EQUATORIAL INSTRUMENT 

Observation with the equatorial we shall find especially useful 
in getting exact positions of the moon, since it is available at any 
time when the moon is above the horizon, and after sunset we can 
always find some bright star sufficiently near to afford a fairly 
accurate value of its place. 

It is often inconvenient to observe the moon by the more accurate 
method which is described in Chapter VI, that of meridian observa- 
tions, which is confined to a short interval of one or two minutes 
each day, and is often interfered with by clouds passing at the 
critical moment, although nine-tenths of the whole day may be 
suitable for observations made out of the meridian. Moreover, 
until the moon is several days old, it is too faint for observation at 
its meridian passage. It is, therefore, upon the equatorial that we 
shall mainly rely for the determination of the moon's motion, as 
well as for many observations of the planets out of the meridian. 

Although it is far more convenient to find the right ascension and 
declination of the sun by the method of the following chapter, at 
least a few positions should be found by observations with the 
equatorial and plotted on the globe. The result will be to show 
that the path of the sun is very exactly a great circle fixed on the 
sphere or so nearly fixed that some years of observation with the 
most refined instruments are necessary to detect any change in its 
position among the stars, although a much shorter time even would 
serve to show the slow change of its intersection with the celestial 
equator due to precession. 

This great circle is called the ecliptic, and its position is shown on 
the map which we have used for plotting our first moon observation. 

Three months will give a sufficient arc of this circle to enable us 
to determine with some accuracy its position with respect to the 
equator, its inclination to the latter, and their points of intersection ; 



MOTION OF THE MOON AND SUN AMONG THE STARS 71 

if possible, observations should, however, be continued throughout 
the year which the sun requires to complete its circuit, so that the 
variability of its motion may be observed, most of the work, how- 
ever, being done with the meridian circle. 

The sun's diameter should occasionally be measured to determine 
the points at which it is nearest to and farthest from the earth. 



CHAPTER VI 
MERIDIAN OBSERVATIONS 

We have now arrived at a point where we can see what are the 
desirable conditions for making observations as accurately as possible 
of the position of a heavenly body. To adjust the equatorial instru- 
ment so that its axis lies in the meridian and at the proper inclina- 
tion, and to keep it so adjusted, is a matter of some difficulty. In 
the last chapter we have shown how, by observing an unknown body 
in a certain fixed position of the instrument, and later a body whose 
right ascension and declination are known in as nearly as possible 
the same position of the instrument, we lessen the effect of the 
instrumental errors. We made our observation of the sun shortly 
before sunset, so that the interval between this observation And that 
of the comparison star should be as short as possible. If, however, 
the rate of the clock can be relied upon, there is no reason why the 
observation should not be made when the sun is on the meridian, 
the interval of time required to connect it with stars in that case 
being not necessarily more than eight or nine hours in the most 
extreme case ; and the comparative ease with which an instrument 
may be constructed so that it shall be at all observations exactly in 
the meridian, and the possibility of constructing very accurate time- 
pieces, has determined the use of such instruments for all the more 
precise observations in astronomy, such as fix the positions of the 
fundamental stars and the vernal equinox on the celestial sphere. 

The equatorial instrument may be used for this purpose by clamp- 
ing it in such a position that the reading of the hour-circle is 0°, in 
which case the declination axis is horizontal east and west, and 
when the telescope is moved about its axis it always lies in the 
plane of the meridian. If, with the instrument so adjusted, we 
observe the sun at the time of its meridian passage, we may find 
its declination by reading the declination circle, and its right ascen- 
sion by noting the interval which elapses before the meridian transit 

72 



MERIDIAN OBSERVATIONS 



73 



of some known star after nightfall, free from any error involved in 
reading the hour-circle. As before, a star should be chosen at 
nearly the same declination, so that the interval of time may be 
very nearly equal to the difference in right ascension between the 
sun and the star, even if the instrument is not very exactly in the 
meridian. Observation of several different stars will enable us to 
determine whether the instrument actually does describe the plane 
of the meridian as it is rotated about the horizontal axis (see Chapter 
VIII) ; and by the observation of stars near the pole, as described 
on page 81, we may determine whether the declination circle reads 
exactly 0° when the telescope points to the equator, as should be 
the case. 

THE MERIDIAN CIRCLE 

An instrument which is to be used in this manner, however, is 
not usually so constructed that it can be pointed at any point in the 
heavens. Thus, it is un- 
necessary that it should 
consist of so many moving 
parts as the equatorial in- 
strument, and steadiness, 
strength, and ease of ma- 
nipulation are very much 
increased by constructing 
it as shown in Fig. 40, 
which represents a very 
small instrument built on 
the plan of the meridian 
circle of the fixed observ- 
atory. The strong hori- 
zontal axis revolves in 
two Y's, which are set 
in strong supports in an 
east and west line. The 
axis is enlarged towards fig. 40 

the center, and through the center passes at right angles the 
telescope tube. The axis carries at one end a graduated circle 




74 LABORATORY ASTRONOMY 

perpendicular to the axis of rotation. If the axis of the telescope 
is perpendicular to the axis of rotation, and if the latter is adjusted 
horizontally east and west, the telescope may be brought into any 
position of the meridian plane, but must always be directed to some 
point of the latter. A pointer attached to the support marks the 
zero of the vertical circle when the telescope points to the zenith, 
and if the telescope be pointed to a star at the time of its meridian 
passage, the angle as read off on the circle is the zenith distance of 
the star ; while the time of the star's meridian passage by a clock 
giving true sidereal time is its right ascension. If the latitude of 
the place of observation is known, the star's declination is deter- 
mined by the fact that the zenith distance plus the declination of 
any body equals the latitude (see page 81). At first the latitude 
may be used as determined by the sun observation of Chapter I, or 
from a good map showing the place of observation, but ultimately 
its value should be determined with the meridian circle itself. 

LEVEL ADJUSTMENT 

We will now proceed to show how to make the necessary adjust- 
ments for placing the telescope so that it may move in the plane of 
the meridian. 

Place the instrument on its pier and bring the Y's as nearly as 
possible into an east and west line. If the pier is the same that 
has been used in the previous work, this may be done by bringing 
the telescope into the meridian which has been determined by the 
method of equal altitudes. 

The axis must first be brought into a horizontal line, making use 
for this purpose of the striding level (Fig. 41), which is a necessary 
auxiliary of this instrument. This is a glass tube nearly but not 
quite cylindrical, ground inside to such a shape that a plane passing 
through its axis, CD, cuts the wall in an arc, AB, of a circle whose 
center is at 0. In this tube is hermetically sealed a very mobile 
liquid in sufficient quantity nearly but not quite to fill it — the 
space remaining, called the "bubble," always occupying the top 
of the tube. When CD is horizontal, the bubble rests in the 
middle of the tube with its ends, of course, at equal distances from 



MERIDIAN OBSERVATIONS 



75 



the middle; the tube is graduated so that this distance may be 
measured, the numbering of the graduations usually increasing in 
both directions from the center of the tube. If the radius of the 
arc is 14.3 feet, a length of 3 inches of this arc will be equal to 

about 1°, since the arc of 1° in any circle is about — — of the radius ; 



iji.o 



1 inch of the arc will then be about 20', and 0.05 inch 1'. These 
are about the actual values for the level used with the instrument 




V 



im&^h 



A MM 


1 


^s9in 




i 

> k 


1 

1 

1 / 
1 / 
1 / 

1 / 


\ 
1 


1 / 
, / 


\ 

\ 
\ 
\ 


1 ' 

1 / 

/ 


\ 


1 



Fig. 41 



shown in Fig. 32, the scale divisions being about ^V °f an mcn 
apart and therefore corresponding to an arc of 1'. 

If the line CD is inclined at an angle of 1' to the horizontal 
line by raising the end A, the center of curvature will be dis- 
placed toward the left, and the level will have the same inclina- 
tion as if the whole tube had been turned to the right about the 
point through an angle of 1'; and the highest point of the arc, 
which is always directly above 0, is now -^ of an inch from the 
middle toward A. Since the bubble always rests at the highest point 
of the arc, it follows that its ends will each be moved toward A by 
one division ; if, for instance, the readings of the ends are 5 and 5 
when CD is horizontal, they will be 6 and 4 when CD is inclined 



76 LABOKATOKY ASTRONOMY 

by V, and evidently 7 and 3 when CD is inclined 2', etc., the incli- 
nation in minutes of arc being one-half the difference of the readings 

A B 

of the ends of the bubble, or — - — if A and B represent the 

readings of the ends of the bubble in each case. If the reading of 
B is greater, the end A is depressed by one-half the difference of 
the readings ; and the above expression applies to both cases if we 
agree that it shall always denote the elevation of A, a negative value 

of — - — indicating depression of A. 



REVERSAL OF THE LEVEL 

The level tube is attached to a frame (Fig. 40) resting on two stiff 
legs terminating in Y's, which are of the same shape and size as 
those in which the axis of the meridian circle rests, the axis of the 
level tube being adjusted as nearly as possible parallel to the line 
joining the Y's. It is difficult to insure this condition, but if it is 
not exactly fulfilled, the horizontality of the axis may still be deter- 
mined by placing the level on the axis, and determining the value 

A jg 

— - — , and then turning it end for end, and again reading the value ; 

for if the end A is high by the same amount in each case, the axis 
is obviously horizontal, and the measured angle of inclination is due 
to the fact that the leg of the level adjacent to A is longer than the 
other leg. The practical rule is to read the west and east ends in 

each position. If these readings are W 1 E± W 2 E 2 , — ^r — - is the 

elevation of the west end according to the first observation, and 
W 2 -E 2 



at the second. If the leg which is west at the first 
Z 

observation is too long, the first observation gives a value for the 
elevation of the west end too great, and the second a value too 
small by the same amount ; and the average of the two values 

— — — - and — ^—t — - gives the true value of the inclination of the 
axis. 



MERIDIAN OBSERVATIONS 77 

It is usual to write this - — , — and to record 

4 

the observations in the following form : 

Wi Ei 

W 2 E 2 



W 1 + W 2 E x + E 2 

Subtract the second sum from the first and divide by 4. This 
gives a positive value if the west end is high, and the axis may 
be made horizontal by turning the leveling screw so as to make 
the level bubble move through the proper number of divisions. 
The level should be again determined in the same way, and the 
axis is level when 

(W 1 + W 2 )-(E 1 + E fi ) = 0. 

The following record of level observation made Feb. 26.3, 1900, 
conforms to the above scheme : 



w 


E 


2 

H 


2 


-l 




— £ division 


= 15" 



The west end being too low, the screw was turned so as to raise it 
enough to move the bubble J division toward the west, the level 
remaining on the axis during the adjustment and watched as the 
screw was turned ; the readings were then as follows : 



2± 


If 


4 


4 




And the axis was truly level, since ( W x + W 2 ) — (E x -j- E 2 ) = 0. 



78 LABORATORY ASTRONOMY 



COLLIMATION ADJUSTMENT 



The line of collimation of the telescope is the line drawn from 
the center of the lens to the wires that cross in the center of the 
field. When the telescope is " pointed " or " set " upon a star, the 
image of the star falls upon the point where these wires cross, and 
when the instrument is correctly adjusted the line of collimation is 
perpendicular to the axis of rotation, so that the line of collimation 
cuts the celestial sphere in a great circle as the telescope turns upon 
its axis. 

To make this adjustment, point the telescope exactly upon any 
well-defined distant point, — the meridian mark will, of course, be 
chosen if it has been located, — : then remove the axis from its Y's 
and replace it after turning it end for end ; if the telescope is still 
set on the mark in the second position, the adjustment is correct; 
otherwise move the wire halfway toward the mark by means of 
the screws a, a (Fig. 40). Set again upon the mark by moving 
the screws in the eyepiece tube ; reverse the axis again, and thus 
continue until the telescope points exactly upon the mark in both 
positions of the axis. 

If the adjustments for level and collimation are properly made, 
the intersection of the wires in the center of the field of view will 
appear to describe a vertical circle, that is, a great circle through 
the zenith, as the instrument is turned on its axis. The final 
adjustment consists in bringing this circle to coincide with the 
meridian, but for this we must have recourse to observations of 
stars. 

AZIMUTH ADJUSTMENT 

The simplest method is to observe the time of transit by a sidereal 
clock of a circumpolar star at its upper transit, and again, 12 hours 
later, at its transit below the pole ; if the interval is exactly 12 
hours, the adjustment is correct ; if the interval is less than 12 
hours, the telescope evidently points west of the pole, and the west 
end of the rotation axis must be moved toward the north. This is 
done by the screws a, a (Fig. 40), the fraction of a turn being noted ; 



MERIDIAN OBSERVATIONS 79 

the observation is repeated upon the following night, and by com- 
paring the change which has been produced by moving the screws, 
the further alteration required is readily estimated. On Feb. 2G, 
1900, the lower transit of e Ursse Majoris was observed at 4 h 58 m 
12 s , and the upper transit at 16 h 53 m 32 s ; the times were taken by 
a sidereal clock and have been corrected for its error ; the interval 
being ll h 55 m 40 s , it is evident that the telescope pointed to the 
east of the meridian, the arc of the star's diurnal path between 
the lower and upper transits lying to the east of the meridian and 
being less than 12 h by 4 ,n 20 s or 260 s . 

To correct the error, the west end of the axis was moved toward 
the south by turning the adjusting screws through one-quarter of 
a turn. On the following day the observations were repeated as 
follows : 

Feb. 27.25, lower transit 4 h 54 m 45 s ; Feb. 27.75, upper transit 
16 h 54 m 28 s ; the eastern arc was still too small, but the error had 
been reduced to 17 s , and required a further correction of 5 y n of a 
quarter turn of the screws, which were therefore turned through 
about 6° in the same direction as before, and the instrument was 
thus brought very closely into the meridian. 

This method can only be used with small instruments when the 
night is more than 12 hours long ; but it is the only independent 
method ; it requires that the rate of the clock shall be known 
between the two observations, and it requires observations at in- 
convenient times. A more convenient method is always used in 
practice, but requires an accurate knowledge of the right ascensions 
of a considerable number of stars in the neighborhood of the pole. 

It has been stated that it is often inconvenient to observe the moon 
when on the meridian, but with this exception all the fundamental 
observations of astronomy are now made with meridian instruments 
on account of the simplicity and permanence of the necessary adjust- 
ments. A body observed on the meridian is also at its greatest 
altitude and least affected by atmospheric disturbances, which often 
interfere with the observation of bodies near the horizon. 



80 LABORATORY ASTRONOMY 

DETERMINATION OF DECLINATIONS WITH THE 
MERIDIAN CIRCLE 

The circle of the meridian instrument may be used to determine 
the declination of a star in two ways, of which that now described 
is perhaps the most obvious, but also the least convenient. 

If the reading of the circle is known when the telescope is pointed 
at the pole, the angle through which the telescope must be moved to 
point upon any star, that is, the polar distance of the star, is the 
difference between this value and the circle reading when the tele- 
scope is pointed at the star ; this angle is 90° — the star's declination; 
if the star is on the equator, the angle is 90°; and if the star is 
south of the equator, the angle is greater than 90° by an amount 
equal to the declination of the star ; if we consider the declination a 
negative quantity for a star south of the equator, the value 90° — 8 
represents the polar distance in all cases. 

To determine the reading of the "polar point'' we may set the 
telescope upon a circumpolar star at its " upper culmination " and 
read the circle, and again, 12 hours later, set on the same star at its 
" lower culmination," the mean of the two readings is the reading 
of the polar point. The effect of refraction may be neglected with 
our small instruments without causing an error of J^ of a degree 
at any place in the United States if we restrict ourselves to stars 
within 10° of the pole, or the circle readings may be corrected by 
a refraction table. Immediately after making this determination it 
is advisable to make a setting on the meridian mark and note the 
reading ; this point may thereafter be used as a reference point from 
which the reading of the polar point may be at any time determined 
if the meridian mark has not in the mean time changed its position. 

Better still, the observation of the polar point may be combined 
with a determination of the circle reading when the telescope points 
at the zenith, by one of the methods to be described later; the 
difference of the readings in this case is obviously equal to the 
co-latitude, and such an observation constitutes an "absolute deter- 
mination of the latitude," that is, a determination made without 
reference to observations made at any other place. When the lati- 
tude has once been satisfactorily determined, the observations of 



MERIDIAN OBSERVATIONS 



81 



the declinations of stars can be made to depend upon determinations 
of the zenith point by means of the fact that for a body on the meridian 

Declination = Latitude — Zenith Distance, 

latitude and declinations being reckoned positive northward from 
the plane of the equator, and zenith distance positive southward 
from the zenith. The proof of this relation is left to the student 
as well as the interpretation of the result when the observation is 
made at the transit below the pole. 

At the time of observing the transits of e Ursae Majoris described 
on page 79 the following readings of the circle were made when 
the star was in the center of the field. Each of these observations 
consists of two readings : one of the index A on the south end of 
a horizontal bar fixed to the supports of the axis, and the other 
of an index B at the other extremity of the bar, as nearly as 
possible half a circumference from A. An angle given by the 
mean of two readings made in this manner is free from the " error 
of eccentricity," which affects readings by a single index in case 
the center of the graduated circle does not exactly coincide with 
the axis about which it is turned between the two observations. 



Date 


A 


B 


Mean 


February 26.25 . . . . 
26.75 .... 
27.25 .... 
27.75 .... 


55°. 45 
39 .95 
55 .45 
39 .95 


55°. 35 
39 .85 
55 .35 
39 .85 


55°. 40 
39 .90 
55 .40 
39 .90 



Hence the reading when the instrument was pointed at the pole 
55°.40 + 39°.90 



was 



= 47°.65. 



Evidently the polar distance of the star was 



55°.40-39°.90 



= 7°.75, 



and its declination 82°.25; and we have thus obtained an "inde- 
pendent " or " absolute " determination of the declination of e Ursse 
Majoris ; that is, a determination independent of the work of other 
observers, and only dependent on the accuracy of our circle and of 
our observations. 



S2 LABORATORY ASTRONOMY 

The circle was known to be adjusted so that the reading of the 
zenith was very exactly zero, hence the latitude of the place of 
observation was 42°.35. The exact agreement of these observations 
indicates that the magnifying power of the telescope was such that 
it could be set more accurately than the circle could be read, and 
not that the results are reliable to a hundredth of a degree. 

For convenience in recovering the zenith reading, in case the 
adjustment of the circle should be disturbed, the zenith distance 
of a meridian mark was measured repeatedly, the result showing 
that its polar distance was 137°.47, and this was used to check the 
polar reading in later observations upon stars when it was impossible 
to get observations of the same star above and below the pole. 

Another method of making absolute determinations of the latitude 
with the meridian circle is to observe the zenith distance of the sun 
at the solstices ; the means of these values being the zenith distance 
of the equator, which is equal to the latitude. This observation, 
however, is subject to considerable uncertainty on account of the 
difference in atmospheric conditions at the summer and winter 
solstice, and to great inconvenience on account of the lapse of 
time; it is, however, of course, the means upon which we must 
rely for the accurate determination of the obliquity of the ecliptic, 
one of the fundamental quantities of astronomy. 

For the use that we shall make of the meridian circle, it will 
probably be most convenient to make a careful determination of 
the polar distance of the meridian mark, and use this habitually 
as a point for reference. 

PROGRAM OF WORK WITH THE MERIDIAN CIRCLE 

Work with the meridian circle should at first consist of reobser- 
vation of all the stars which have been previously observed with 
the equatorial, except those which are west of the meridian after 
nightfall and cannot be observed for six months. Attention should 
be given to gathering a list of stars within 15° or 20° of the pole 
for the purpose of quickly setting the instrument in the meridian 
by the method of page 120. The sun should be observed at least 
once a week and its place plotted on the globe, and many stars 



MERIDIAN OBSERVATIONS 83 

in the neighborhood of the moon's path to form a basis for finding 
the moon's place by differential observations, of course, also the 
moon itself, the planets and a comet, if any of sufficient bright- 
ness appears. In this way, by observing a few stars each night, 
a great amount of material may be stored for future use. 

Especial attention should be given to getting a good number of 
observations of stars near the equator, so that fairly accurate values 
of their differences of right ascension may be obtained, and at the 
first opportunity the absolute right ascension of one of their num- 
ber must be determined in order that thus the places of all may be 
known. The results may be best recorded by making a list of 
their right ascensions referred to an assumed vernal equinox. Thus, 
the observations discussed on page 52 show that a Pegasi precedes 
y Pegasi by 17°.03 = l h 8 m 7 s , or, in other words, follows it by 
22 h 51 m 53 s ; and if the right ascension of y Pegasi referred to the 
assumed equinox is h 8 m , that of a Pegasi is 22 h 59 m 53 s . If 
in the course of the year observation shows that the true right 
ascension of y Pegasi is h 8 m 5 s , it is evident that the true value 
for a Pegasi is 22 h 59 m 58 s , and that the right ascension of all stars 
referred to the assumed equinox by comparison with y Pegasi must 
be increased by 5 s . 

DETERMINATION OF THE EQUINOX 

An opportunity for observing the absolute right ascension of the 
zero star, which is often called a " determination of the equinox," 
occurs about the middle of March and September. 

If the course is begun in September, it will be well to make this 
determination with the help of more experienced observers, even 
before the nature and object of the measures are understood. 

The observation consists in determining the difference of right 
ascension of some star from the sun at the instant when the latter 
crosses the equator, for at that time it is either at the vernal or 
autumnal equinox, and its right ascension is in the one case hours 
and in the other 12 hours. 

If a meridian observation of the sun's altitude shows that the sun 
is exactly on the equator at meridian passage, and the time of transit 



84 LABORATORY ASTRONOMY 

is noted by a sidereal clock, and as soon as it is sufficiently dark the 
transit of a star is observed, the difference of the times is the absolute 
right ascension of the star if the observation is made at the vernal 
equinox, or equals the right ascension of the star minus 12 h if the 
observation is made at the autumnal equinox. 

Inasmuch as the meridian of the observer will rarely be that 
one on which the sun happens to be as it crosses the equator, we 
must make observations on the day before and the day after the 
equinox, thus getting the difference of right ascension of the star 
from the sun at noon on both days. The declination of the sun 
being also measured at these two times, a simple interpolation gives 
the time at which the sun crossed the equator, and this time being 
known, another simple interpolation between the differences of right 
ascension at the two noons gives the difference of right ascension 
of the sun and star at the time when the sun was at the equinox, 
which is the star's absolute right ascension. 

The first interpolation assumes that the sun's declination changes 
uniformly with the time, and the second that its right ascension 
changes uniformly with the time. 

Observations should extend over a period of a week before and a 
week after the equinox to test the truth of these assumptions. 

In observing the sun, a shade of colored or smoked glass may be 
placed over the eyepiece, or the eyepiece may be drawn out as in 
the method of observation described on page 37, and the screen 
held in such a position that the cross-wires are sharply focused 
upon it. As the image of the sun enters the field it should be 
adjusted by moving the telescope slightly north or south till the 
horizontal wire passes through the center of the disk, and as the 
latter advances, the time should be noted when the preceding and 
following limbs cross the vertical wire, as well as the time when 
the vertical wire bisects the disk ; at the instant of transit the disk 
should be neatly divided into four equal divisions, a very small 
deviation from this condition being quite perceptible to the eye. 



MERIDIAN OBSERVATIONS 



85 



THE AUTUMNAL EQUINOX OF 1899 

The following table gives the details of observations taken at 
the autumnal equinox of 1899 for the purpose of determining the 
equinox. 

The latitude of the place of observation was 42°.5, and the declina- 
tions given in the last column are calculated by subtracting the zenith 
distance in each case from this quantity, as explained on page 81. 



Date 


Object 


Time of Transit 


Zen. Dist. 


Decl. 




Sept. 22 


Sun 


12h 


m 


2 s .O 


S42°.2 


+ 0°.3 






rj Serpentis . . 


18 


18 


22.6 


45.4 


- 2 .9 






X Sagittarii . . 


18 


24 


2.4 


67 .95 


- 25 .45 






Vega .... 


18 


35 


44.5 


3.87 


+ 38 .63 






Altair .... 


19 


48 


7.6 


33.98 


+ 8.52 




Sept. 23 


Sun 


12 


3 


45.1 


42 .62 


- .12 






t) Serpentis . . 


18 


18 


20.1 


45.4 


- 2 .9 






X Sagittarii . . 


18 


23 


57.3 


67 .97 


-25.47 






Vega .... 


18 


35 


42.6 


3 .85 


+ 38 .65 






Altair .... 


19 


48 


1.5 









The intervals between the observed times of transit of each star 
on the two different dates range from 23 h 59 m 53 8 .9 to 23 h 59 m 58M, 
showing that the clock was losing about 4 8 daily, a quantity so 
small that for our purpose it may be neglected. 

Observations of the sun made on different dates between Sep- 
tember 18 and September 23, but not here recorded, showed that 
its right ascension and declination were changing uniformly at the 
rate of about 3 m 45 s and 0°.39 per day. The table above shows 
that from September 22 to September 23 the rates were 3 m 43 8 .1 
(or, allowing for clock rate, about 3 m 39 s ) and 0°.42 per day, and 
the latter value we shall use to determine the time of the equinox, 
as follows : 

At noon September 22, or September 22 d .O, as it is expressed by 
astronomers, the sun's declination was + 0°.3, and September 23.0 



86 



LABORATORY ASTRONOMY 



its declination was — 0°.12. Hence its declination was 0° Septem- 
ber 22£§, or September 22 d .714. It was at that time, as exactly as 
our observations can show, at the autumnal equinox, and its right 
ascension was 12 h m 8 . 

Since -q Serpentis followed it to the meridian 6 h 18 m 20 8 .6, that 
quantity is the difference between the right ascension of the star 
and that of the sun September 22.0. Similarly the difference of 
right ascension of sun and star September 23.0 was 6 h 14 m 35 s .O ; 
that is, it was 3 m 45 s .6 less than at the previous date. Assuming 
this change to be uniform, the difference of right ascension of sun 
and star at the moment of the equinox on September 22 d .714 was 
0.714 X 3 m 45 s .6, or 2 m 41M less than on September 22.0; that is, 
it was 6 h 15 m 39 8 .5, and since the right ascension of the sun Sep- 
tember 22.714 was 12 h m s , the right ascension of rj Serpentis was 
18 h 15 m 39 8 .5. 

The following table gives the data from which the "absolute 
right ascensions " of the four stars are thus determined. In the 
last column are the declinations, which are the means obtained from 
several observations between September 14 and September 23. 



Star 


R.A. of Stab minus R.A. of Sun 


Star's 


Sept. 22.0 


Sept. 23.0 


Sept. 22.714 


R.A. Decl. 


rj Serpentis 


Qh igm 20s. 6 


6h 14m 35s. o 


6 h 15m 39s. 5 


18* 15™ 39*. 5 


- 2°. 89 


\ Sagittarii 


6 24 0.4 


6 20 12.2 


6 21 17.4 


18 21 17.4 


-25 .48 


Vega 


6 35 42.5 


6 31 57.5 


6 33 1 .8 


18 33 1.8 


+38 .65 


Altair 


7 48 5.6 


7 44 16.4 


7 45 21.9 


19 45 21 .9 


+ 8.59 



The measurements upon which the above results depend are of 
two kinds : observed clock times, which are liable to errors of a 
very few seconds, so that the differences of right ascension may be 
assumed to be correct within perhaps 4 s ; and measures of the sun's 
declination, which with the greatest care may be in error at least 
0°.05 on any given date. 

It is quite within the bounds of probability, for instance, that 
the sun's decimation was -f- 0°.25 on September 22.0 and — 0°.17 



MERIDIAN OBSERVATIONS 



87 



on September 23.0; and recomputing with these values, the date 
of the equinox was September 22|#, or September 22 d .595, and 
the right ascensions of the stars 18 h 16 m 6 8 .4, 18 h 21 m 44 8 .6, 
18 h 33 m 28 8 .6, 18 h 45 m 49 8 .2 ; that is, the uncertainty of the equinox 
is 0.12 days and of the right ascensions about 27 s , although the 
relative right ascension is altered only by a fraction of a second 
in each case. It is thus evident that the accuracy of the right 
ascensions depends chiefly upon the accuracy with which the sun's 
declination can be measured. 

THE AUTUMNAL EQUINOX OF 1900 

In order to increase the accuracy of determination of declination, 
a new circle reading to minutes of arc was substituted for that 
used for the observations of the equinox in 1899, and the observa- 
tions were repeated at the same place in 1900. The weather con- 
ditions were unfavorable, so that only the following observations 
could be made. 



Date 



Sept. 22 



Sept. 23 



Object 



Sun . 

Vega . 
Altair 

Sun . 
Altair 



Time of Transit 



11* 


58'" 


44«.8 


S42° 


11'.5 


18 


35 


27.0 


Q 


49.0 


19 


47 


49.0 


33 


51.0 


12 


3 


1 .5 


42 


33.1 


19 


48 


35.0 


33 


54.0 



Zen. Dist. 



Decl. 



+ 0° 18'. 5 
+ 38 41.0 
+ 8 39.0 



- 

+ 8 



3.1 
36.0 



From these data, by the same method as before, the date of the 
equinox is found to be September 22^f;f , or September 22.8565. 
If each declination of the sun is accurate to 1', the result may be 
in error by gfrs days, or about .09 day ; the actual error is probably 
less than half this amount, and the concluded right ascensions 
probably within 10 s of the true values. 

The observed times of Altair on the two dates show that the 
clock was gaining 46 s daily, since the true sidereal time of transit, 



88 



LABORATORY ASTRONOMY 



being equal to the star's right ascension, is the same on both nights. 
This rate is so large that it cannot be neglected as in the discus- 
sion of the result for 1899. 

If the clock correction At (see page 60) at the time of the sun's 
transit, September 22, be assumed s and the gaining rate 46 s per 
day, or 1 8 .916 per hour, the corrections for Vega and Altair Sep- 
tember 22 were — 12 s . 6 and — 14.9, and for the sun and Altair 
September 23 were — 45.9 and — 61 8 .0. The times obtained by 
applying these corrections are said to be "corrected for rate of 
the clock to the epoch September 22.0." 

In this manner the times, as they would have been observed with 
a clock having an exact sidereal rate, are found to be : 





September 22 


September 23 


Time 

u 
u 


5 of transit of the Sun . . . 
" " Vega .... 
" " Altair .... 


Hh 58m 44s. 8 

18 35 15 .4 

19 47 34.1 


12h 2m 15 s . 6 
19 47 34 .0 



Hence Altair followed the sun 



September 22.0 


7h 48m 4QS.3 


23.0 


7 45 18 .4 


22.856 


7 45 48 .8 



and the right ascension of Altair was 19 h 45 m 48\8 ; since Vega pre- 
cedes Altair by l h 22 m 18 s .7, its right ascension was 18 h 33 m 30M. 

In 1899 the difference of right ascension of the two stars was 
l h 22 m 20M, but the right ascensions of 1900 are greater by 28 8 .3 
and 26 s .7 than those of 1899. 

If we assume the later determination to be absolutely correct, 

we must regard the earlier as having placed the equinox farther 

toward the east among the stars than its true place, so that right 

ascensions referred to the equinox observed in 1899 are too small. 

We may say that the observations of 1900 indicate a correction of 

— 27 8 .5 to the " equinox of our little catalogue of four stars " ; that 

is, a correction of -f- 27 s .5 to all their right ascensions as determined 

in 1899. . , n 

Lor 0. 



MERIDIAN OBSERVATIONS 89 

Applying these corrections, their right ascensions become for 

7] Serpentis 18 h 1G™ 7 s .O 

X Sagittarii 18 21 44 .9 

Vega 18 33 29 .3 

Altair 19 45 49 .4 

Since the later observations were made with an instrument 
giving more accurate values of the declination, it is probable that 
their results are more nearly correct. The clock rate was neglected 
in the first observations, and the effects of precession, parallax, and 
refraction in both series, following out the principle that no correc- 
tions will be made until observations shall show their necessity. 

The effect of refraction is to delay the autumnal equinox about 
an hour, and hence to decrease the right ascensions of the stars 
by about 10 s . At the vernal equinox, however, refraction hastens 
the equinox an hour and increases the right ascensions by 10 8 ; 
its effect may be shown by observations at the two equinoxes of 
the same year and eliminated by their combination. Parallax 
hastens the autumnal and delays the vernal equinox by about 8 m , 
thus affecting right ascensions by a little more than I s , the mean 
of observations at the two equinoxes being free from error from 
this source. The effect of precession will be manifest in less 
than ten years with an instrument like that used in the above 
observations of 1900. 

By comparing the equinox of September 22.714 ± 0.12, 1899, and 
September 22.856 ± .09, 1900, the length of the tropical year is 
found to be 365°.142, but may lie between 364.93 and 365.35 as far 
as our observations can surely determine. Since refraction delays 
the vernal and hastens the autumnal equinox by nearly the same 
amount (about an hour) in each case, it has no effect upon the 
length of the year. As the greatest error to be feared with our 
improved instrument is less than 0.1 day, the length of ten or one 
hundred years may be determined with less than twice that error, 
in those periods the length of the year may be determined within 
0.02 and .002 day, respectively. 

With the best modern instrument used to the greatest advantage, 
the sun's declination may be determined near the equinox within 



90 LABORATORY ASTRONOMY 

0".5, and hence the time of the equinox within 30 s and right 
ascensions within S .08. A single tropical year may be measured 
with an error of less than l m . 



We have now explained the methods by which it is possible to 
fix the places of the sun, moon, and stars at different times and 
thus to obtain data from which their apparent motions about the 
earth may be studied and theories formed from which their future 
places may be predicted. More or less complete accounts of these 
theories are to be found in all works on descriptive astronomy, 
and the predictions derived from them are published for three 
years in advance by several governments for the use of navigators 
and astronomers. Such a publication is the American Ephemeris 
and Nautical Almanac, of which it will be convenient to give some 
account before taking up the motions of the planets. 

The apparent motions of the planets are less simple than those 
of the sun, moon, and stars, which at all times seem to move about 
the earth as a center with approximately uniform velocities. The 
planets, it is true, in the long run continually move like the sun 
and moon around the heavenly sphere toward the east, but their 
velocities are variable within wide limits and at certain times are 
even reversed, so that they move in the opposite direction or 
"retrograde" among the stars. 

For this reason a longer period of observation is necessary to 
determine their motions than can be given by the individual student. 
We may, however, regard the nautical almanacs of past years as 
predictions that have been verified, and they stand for us as an 
accredited set of exceptionally accurate observations from which 
we may draw material to combine with the results of our own 
observations. 



)UN 15 1901 



LIBRARY OF CONGRESS 




003 538 500 9 



